How does acceleration affect time dilation and length contraction? I just studied SR and I understand how velocity affects time and length; how do they evolve during the process of reaching a certain velocity?
 A: At any instant, the time dilation and the length contraction of a moving object just depend on its instantaneous velocity relative to the observer, not on its acceleration.
If the particle is accelerating and you want to know, for example, the time that elapses on “its clock” over a finite interval, then you have to integrate over each infinitesimal interval of time that elapses for it. In doing so, you will take its acceleration into account because you have to use its varying instantaneous velocity.
A: For the purposes of your question, acceleration is the process of going from one inertial frame to another.
The differences you see between being in one frame and the other occur during acceleration.
If you are 'looking at' events in a chosen inertial frame as you accelerate, you will see lengths changing - decreasing or increasing as the relative speed increases or decreases.  Different other inertial frames will exhibit this to different extents depending on how they are moving.
Similarly you will see clocks in other inertial frames speeding back up towards the 'normal' speed of your own clocks, or slowing down more as the time dilation effect changes for each one.  Again different 'other' inertial frames will show this effect to different extents.
All of this is pretty boring, features of your perception of other frames rather than something you can associate specifically when you start your acceleration with your acceleration.
What is much, much more interesting is the relativity of simultaneity.  In your final inertial frame, when you look at the clocks in the inertial frame you left, the clocks 'ahead' of you in that frame show later times than those behind you.  You are looking into the relative future of your initial frame, and if you look behind you, you see events in its relative past.  The time differences the clocks show are proportional to their distance away from you and the relative velocity between the frames - your speed change.  
The proportionality is to the distance they see in their frame.
If you increase your relative speed again, to a higher value, three things happen.  First, all the clocks in that frame appear too slow down a bit.  That doesn't change the time differences you see between them.
Second, they get closer together in the direction of travel, so the proportionality with distance that you see increases for that reason.
But third, the relativity of simultaneity effect increases, so they all advance by an amount proportional to your increase in speed, and because of the way Special Relativity works, proportional to the distance away you see them to be.
The effect is the same for all the inertial frames you choose to look at, including your own, so a clock in your own frame kept a fixed distance (in your frame) from your clock will show by its time difference how much your speed has changed since they were synchronised. Sadly, while you might measure this total change to be as much as you like, the accumulated total speed is somewhat less, and can never exceed light speed.
So the effect of acceleration is it causes all the clocks around you to apparently go faster (or slower) by an amount proportional to your acceleration multiplied by the distance away (in the direction of acceleration) they appear to be.
And yes, at a negative distance from you, depending on their speed in the frame you choose to look at, some will be stopped, and beyond them others will seem to be running backwards.  And yes again, at positive distances from you clocks will apparently be racing into the future faster than your own.
Meanwhile, of course, the people around you in their own inertial frames, are quite unaware of the weird view you have of their worlds.
And all of this comes from careful observation of Einstein's train.  It is the result of the Lorentz transformation as the train speed changes.
And last of all, to open a small window into General Relativity, if the acceleration you feel is a constant 1G the 'leading' clock in your own accelerating frame of reference gains the same amount of time as a higher clock on Earth at the same distance, at least locally.
Relativity can be a real Wonderland.
