Why are particles different sizes? Is it correct in saying that a particles size is it's rest energy, and that particles don't actually have size (in the way you get different size objects)?
What defines what sizes a particle can be? Why do particles have discrete sizes, and there's not a continuous spectrum of particles varying in size?
I ask because I was told that particles mass depends on its size, as the bigger a particle is the more it interacts with the Higgs boson and so the more mass it has. So why are there so few particles with specific masses/sizes? 
 A: 
Is it correct in saying that a particles size is it's rest energy, 

No

and that particles don't actually have size (in the way you get different size objects)?

Not really.

Why do particles have discrete sizes, and there's not a continuous spectrum of particles varying in size?

They don't have discrete sizes.

OK, let me explain:
Fundamental particles are point particles by the Standard Model (String theory gives them a size, which seems to be somewhat correlated with mass, but I'm keeping String theory aside here).
Of course, we have fundamental particles of varying rest energies, so size is not related to mass.
Now, let's look at composite particles. Say, a proton. Protons are made of three quarks, along with some virtual gluons that are being exchanged. The size of the proton is thus the volume in which all its constituent particles are confined.
But, the constituent particles are not confined--there is a very slim chance of finding one of the quarks a few meters away{*}, due to the wave nature of the particles. So, we define size as the volume where you will find all the particles in <some arbitrary percentage> of the time (where that arbit percentage will be somewhat like 99.999%)
The definition need not be "all the particles will be found here x% of the time"; we can have fundamentally similar, but different definitions. For example, as @dmckee noted, the charge radius of a nucleus is where the charge density drops to $e^{-1}$ of the central density.
Similarly, for an atom, you could define size via the boundary of the electron cloud. But, the electron cloud stretches to infinity, so you have to arbitrarily define size via a percentage(or something similar).
In fact, for an atom, going across a period, size decreases as mass increases (but not when going down a group). So, for composite particles, size and mass have no fixed relation.
*I'm not too sure of this, color confinement may prevent the quark from going so far without pair production.
A: The rest energy of a particle determines its mass but not its size.  
Although it is often said that elementary particles are points, the correct statement is to say that they are pointlike, because due to renormalization corrections, the point particles that correspond to the bare fields in the defining Lagrangian become slightly extended.
The size of a particle is determined by how the particle responds to scattering experiments, and therefore is (like the size of a ballon) somewhat context-dependent. 
My theoretical physics FAQ contains two chapters with several entries related to questions of particle size (and shape, which is related), where much more detail can be found:
Chapter A6: The structure of physical objects
   - Does an atom mostly consist of empty space?
   - How do atoms and molecules look like?
   - When is an object macroscopic?
Chapter B2: Photons and Electrons
   - Are electrons pointlike/structureless?
   - The shape of photons and electrons
