Is it heat that causes vibrations on particles or is it the vibrations that causes the heat? I was taught that molecules move more random when its hot. I am just wondering what really is it that causes something to heat up.
 A: I would go for the first version:

[It is] heat that causes vibrations on particles

but more accurately I would rather say:

[It is] heat that is our name for vibrations on particles

Neither "causes" the other, since they are not two different things - one is just a name for the other; one is micro-scale and the other macro-scale (in the case of random motion/vibrations).
A: In addition to those nice answers, I would point out that the words "heat" and "heat up" are examples where the term "heat" is used in very different ways, so one must be careful.  
Generally speaking, "heat" means you have energy in transit, with the additional connotation that the form of the energy is highly randomized, or "thermalized," as the other answers stressed.  But "heat content" is something of a misnomer, because heat is not a state variable, meaning, you cannot tell me the state of the system and thereby know how much heat was put into it to arrive at that state (because, if you start with some systems at very low temperature but various different volumes, you could get them all to arrive at the state you have in mind by putting in different amounts of heat, depending on their history).  
Usually, when the concept of "heat" is used synonymously to something like "random energy content", the implication is it got there at constant volume (i.e., no work to worry about), and then the second law says the internal energy U, which is a state variable, is equal to the heat added Q.  In that case, U will also equal the internal random kinetic energy.  
But even then there is a wrinkle in quantum systems-- U will not necessarily be what we mean by "thermal energy content", because the gas might be showing quantum mechanical effects, like becoming "degenerate."  If it's degenerate, its thermal energy content is way less than its internal energy, and the heat added at constant volume will relate to the latter, not the former.
Also, the term "heat up" is even more ambiguous, because what people usually mean by it is nothing more than "rise in temperature."  But if you include work, you can actually get things to rise in temperature by removing heat from them-- a classic example is the formation of a star, where you have a gas cloud that is always losing heat while it "heats up," in the sense of rising temperature. 
So these terms are tricky in general cases!  But in the simplest case of classical free particles at constant volume, or classical springs, then your picture works fine-- adding heat means raising the internal random kinetic energy, and the presence of random internal energy means you have some concept of "heat content", which is the heat you need to add to reach that state from a very low temperature.
A: The fancy word terminology for this is that the concept of heat supervenes on the concept of molecules moving.
We can think about heat as molecules moving.  We can also think about a lot of non-heat things as molecules moving too.  The kinetic energy of molecules moving is a very low-level concept, so it can be used to describe a lot of things.
The only tricky part of tracking the kinetic energy of molecules is that there's a lot of them, and they're often really disorganized.  Sometimes it makes sense to treat their movement as "random," with some distribution.  When we look at it this way, we call the energy of those random molecular movements "heat."
Supervenence is a really cool term that I borrow from philosophy.  It refers to some high level property which can be completely understood as a lower level property.  The concept of "heat" is a high level property which can be completely understood by looking at the random motion of molecules.  The reverse is not true.  There are movements of molecules which are too ordered to be treated as random motion, so those molecular movements cannot be thought of as "heat."  Only the random movements can be thought of that way.
Lumping all of those random movements together and calling the result "heat" has a lot of nice properties.  For example, it permits us to discuss things like engines, which rely on the properties of gasses exhibiting these random movements, without getting too caught up on what the individual molecules are actually doing.
A: It's both: At the smallest scale, thermal energy is just kinetic energy, the energy of motion. That is, thermal energy (heating up) and kinetic energy are the same thing. 


*

*When molecules vibrate, they're bumping into each other—transferring kinetic energy to other molecules, which sometimes radiate this energy as heat (on a larger scale). Note, too, that their vibrations are an expression of kinetic energy.

*When the environment in which molecules exists heats up, the molecules absorb some of this [kinetic] energy and thus vibrate more, a little bit like knocking one billiard ball with another causes the second to move. 

A: We can't feel individual vibrations of this kind; it would be like trying to feel each individual molecule in a wave as they press (or more accurately exert force or rebound) on your skin. 
We feel the overall sensation of heat, or pressure, which is made up of untold numbers of energy-transferring collisions and vibrations.
What you are asking is a bit like asking if molecules moving in the air cause what we feel as "wind blowing on us", or the wind blowing on us moves the molecules in the air. 
In a computing sense it's a bit like asking if a computer is running the BASIC program "20 print x" or the machine code of an interpreter. 
Depending which level you want to look at it, either - and you choose the way to look at it based on what's useful to your goal of looking at it, because both are valid answers. It isnt that one is "true" and one is "less true".
Physically, the molecules (or other particles) are moving. Overall if their movement transfers energy to us we might feel that and name it "warmth". If they exert a net force on us we might feel that and name it "pressure". 
We do this because that's mostly how, as life forms, our nervous system 'reports' these sensations.
What causes an individual particle to speed up is a different question. But the above seems to be the question you really meant - is it?
