Why does it seem everything I push moves at a constant velocity? I am aware that a constant force causes a constant acceleration but friction can counteract this. However, if I push something across a table, for example, it seems no matter how hard I push, the object travels at a constant velocity, even if I apply more force than the kinetic friction. The object seems to always travel at the same velocity as my hand, does this mean I am not actually applying a constant force?
 A: 
I am aware that a constant force causes a constant acceleration but friction can counteract this.

True. Although any non-zero net force acting on an object causes it to accelerate. The net force does not have to be constant in time for acceleration to happen.

However, if I push something across a table, for example, it seems no matter how hard I push, the object travels at a constant velocity, even if I apply more force than the kinetic friction. 

Well, the object was at rest on the table. Then you pushed it and it started moving. Therefore the velocity of the object changed, and you caused it to accelerate. The object is obviously not traveling at a constant velocity.

The object seems to always travel at the same velocity as my hand, does this mean I am not actually applying a constant force?

You probably are not applying a constant force (or maybe you are. I cannot say without being there and actually measuring the force you apply). But you are for sure applying some force which is causing the object to accelerate. Your hand would then also be accelerating while it is in contact with the object. This doesn't mean the object has a constant velocity though.
A: It's not easy pushing something with by hand with a constant force greater than kinetic friction.
Try using a rubber band and a ruler to pull something across the table with a constant force.  I think if you focus on keeping the rubber band stretched a constant amount while you pull you will notice the object will accelerate.
A: I don't know if you've learned about energy and power yet, but if you have, that leads to a pretty plausible explanation. As you push this object with force $F$ at speed $v$, the power you expend (like the horsepower of a car) is given by $P=Fv$. As you start the object in motion, it's accelerating. But this process of acceleration is limited by the power your body is comfortable supplying. To keep accelerating, you need to supply a force greater than the force of friction $F_f$, which means that $P>F_fv$. You run into a limit at speed $v=P/F_f$.
When it's not physically difficult, like moving a coin across your desk, then I don't think it's true that it moves at constant velocity, unless you choose to make it so. You could choose to flick the coin or something. This makes sense because you're not running into your power limit.
A: You are not pushing the object with the constant force. 
It's not a question of physics but the one of biology. Your brain does not command your muscles to apply the constant force, it commands to move the body part to arrive at a certain position or move at a certain speed by applying whatever force is necessary. The only exception would be applying pressure against the stationary object, but that's not the case here.
A: I actually think this is kind of a great question, but I'd like to focus on one of your statements in particular:
"The object seems to always travel at the same velocity as my hand, does this mean I am not actually applying a constant force?"
If you are pushing an object, this will be true regardless of the force you apply. That's part of the definition of the contact interaction - the object you're pushing will always have the same velocity as your hand. If it had a different velocity, you would no longer be pushing it!
So, in order to make an object accelerate, you need to make your hand accelerate. If you push an object in such a way that your hand moves faster and faster, the object will also be accelerating.
Now, if you do that, you might not be pushing with a constant force, but to me it's very difficult for a human to determine if a force is constant. I would recommend getting a simple force sensor to check that the force you're applying is actually constant.
A: Let us look at the problem of terminal velocity

When an object which is falling under the influence of gravity or subject to some other constant driving force is subject to a resistance or drag force which increases with velocity, it will ultimately reach a maximum velocity where the drag force equals the driving force. This final, constant velocity of motion is called a "terminal velocity", a terminology made popular by skydivers.

Bold mine.
There is no acceleration and the net force is zero. The sky diver falling under terminal velocity has no acceleration, the drag forces and the force of gravity add to zero.
In the case of your hand applying a constant force , if true, it means that the terminal velocity i.e. where friction force and your force add up to zero has been reached.
It needs though a better experiment to see if this is true, with control of the force and the velocity measurement.
A: Adding to the already posted answers, I'd like to encourage you to experiment a little to better grasp the relation between force, acceleration and velocity:
Get yourself a spring balance, to be able to measure the applied force (instead of just feeling it). Use the spring balance to drag some object along an even surface at different speeds. You should find that the applied force only changes, when changing the speed (i.e. while accelerating). The measured force should, however, be the same for different constant speeds (more or less, considering measurement errors), because you only need to counteract the force of friction, which is independent of the actual speed.
Try the experiment with different objects and/or different pairings of materials (object and surface). You should find that mass and the friction coefficient are the only factors that influence the required force to achieve any constant speed.
If you want to take my word on the outcomes, this can suffice as a thought experiment. I'd recommend to get your hands dirty though ;)
A: Experiment: Try moving your mouse (if you use one) across the screen at a constant speed. 
This should feel difficult to achieve, especially if you aren't resting your wrist or elbow on anything. Your brain wants the mouse to move at a constant speed, but your muscles are probably overcorrecting every time you see the cursor speed up or slow down.
The reason it's so hard to match your applied force with kinetic friction is that your body is moving. Joints are bending, so the torque your muscles are applying to your bones changes, therefore even a constant muscular tension would result in a changing movement of the mouse. Or consider the opposite: when you hold a bag of groceries at your chest, it's fairly easy to hold it close to constant (zero) velocity -- the angle between your bicep and forearm is constant, so it's easier for your brain to lock in on the correct muscular force to achieve constant velocity.
A: Other answers have correctly explained that it's very difficult to apply a constant force by hand on an object on a table. However, if we change a bit the experimental setting we can apply a fairly constant force on an object by hand and feel it accelerating.
Part of the problem of pushing an small object on a table is that the forces involved are small and movement is fast. Specially, forces are smaller than the weight of the hand and the arm -making it difficult to appreciate them- and movement is fast enough to make body position change in less time than needed to accurately appreciate force.
I suggest changing the object to a heavier one that could be pushed with a strong force and still accelerate little enough that our body position don't change much while pushing. Then, I would replace the object on your table by a car on a flat surface.
In my experience, when you start pushing a car it accelerates slowly, but if you keep pushing hard (evenly hard) for some seconds it keeps accelerating until you stop pushing to keep the experiment safe.
Of course, force is not exactly constant when pushing a car, but pushing as hard as you can against a car at rest or against a car moving at a few centimetres per second is quite similar. Furthermore, once the car has started moving friction doesn't change much in this velocity range.
A: It's very difficult to apply a constant force to a small object with your hand. Your hand will move at the speed your brain commands it, rather than applying a constant force.
Try pushing something very heavy, such as a boat, car, or a trolley stacked with drinks, stones or another person. Then you will have to push with constant force (the max force you can push with) to get the object moving and will notice the acceleration - it starts off moving very slowly, then builds up speed. Once you do get it moving and assuming the surface is level, you will find you are limited either by friction or your ability to keep up the speed.
A: 
The object seems to always travel at the same velocity as my hand,
  does this mean I am not actually applying a constant force?

Yes it does mean you are not applying a constant force. The force is due to an interaction at the surfaces of your hand and the object, and that interaction depends on how closely in contact and compressed those two surfaces are. The compression depends on the very precise difference in position between your hand and the object.
Let's assume your hand and the object each have constant speeds. Of course the speeds will change later, but this assumption is realistic for thinking about a very short time period.
If the object is moving slower than your hand, then it is getting closer, and if touching both things compress. As this happens the force very rapidly increases.
If the object is moving slower, then the opposite happens, and the force very rapidly decreases.
The result of this is that the force rapidly stabilizes to whatever level is necessary to move the object at the same speed as your hand.
A: It is just because the table is (1) giving enough friction to slow down the object so that it is always touching your hand, but (2) your hand can push with greater force than the friction can act on the object.
To see for (1), imagine if the table is very smooth and you put some soap water on it, and now push a metal ball. Another scenario is to have a lot of tiny metal balls on the floor and you push a box "floating" on the tiny metal balls. The stronger you push, the faster the object will travel. You can even push the object so fast that it zoom forward and your own body cannot catch up with it.
To see for (2) imagine an ant trying to push the object. The ant cannot provide greater force than the friction, so the object cannot even move. Now change the ant to a mouse, and then to a dog, and you can see that when the force is greater, the object can move.
You can also see that with enough force, with that friction, the object can move very fast: if you use some long and strong rubber band and pull them far, and release it and let the rubber band catch the object inside of the rubber band,  you can see that the object zoom forward with a great velocity. Another way is to swing a baseball bat and hit that object, and it will travel fast even when the baseball bat has stopped at the hitting point.
A: You are applying a constant force, but the constant force happens to exactly equal that from kinetic friction, hence the total force, which sets the acceleration the block experiences, is zero and it moves with constant speed, equal to that of your hand.
You see, this kind of experiment is one that, because of its reliance on a human participant - an active element - happens to (inadvertently) introduce questions of human biology into the mixture, not simply pure physics. Your brain is actually subconsciously regulating the amount of force coming from your hand so as to not let the object experience a consistently greater force than kinetic friction because it would then start to "get away" (i.e. accelerate). As your intent is to keep pushing the block, your brain's way of dealing with that is to regulate the force from your hand.
When you start pushing the block, your brain lets you increase the force from your hand until the static friction breaks (i.e. it reaches its maximum and then the object begins to move leading to a shift from the static to kinetic regime), then it soon drops the force from the hand to equal kinetic friction to keep the object "under control". You feel this process as a seeming "pop" when it starts moving. That "pop" is the object briefly "getting away" (i.e. accelerating away from your hand) followed by it slowing down again (as kinetic friction robs it of speed) until your hand moves back into firm contact with it, causing a change in the pressure on your hand throughout the process.
This is why that pure physics experiments should not involve a human in them in this fashion. Another example of this is the "why does it feel like I'm struggling and making 'effort' pressing against an unmoving wall when 'physics' says that the work done by the force generated by my hands in that situation is zero? Clearly there has to be 'work' of some kind being done here!" Again, the answer is in human biology, not physics: because of how they are constructed on a molecular level, your muscles need to continually expend energy simply to keep a constant force even if it is doing no work. On the other hand, a purely passive element like a spring caught between a post and the wall  does not expend energy unless it is expanding, so that the point of force application is moving, just as in line with how physics operates.
