What do gravitons do? Yes, I know gravitons are 'just a theory', but I'm wondering how they theoretically act. Are they raining down on everything with mass to "push" other things towards it? Or do the gravitons form a sort-of "field" that gives everything gravity (like Higgs field)?
 A: From the analogue of simple quantum mechanics Feynman diagrams, the graviton is what is being exchanged for two particles to feel an attractive force. Analogous to the exchange of a photon for the electron to interact with another electron. 


first order feynman diagram electron electron interaction

The analogous diagram for the gravitational interaction of two electrons needs a graviton exchanged. The problem is that this simple analogy breaks down for calculational purposes, in the case of the graviton because the higher order diagrams which in the case of photon exchange are much smaller and vanish in very high orders (using renormalization), for the spin two graviton they diverge (non re-normalizable) leading to nonsense. Thus it is only in principle the analogy can be taken, assuming that the final theoretical quantization of gravity will take care of this .

Or do the gravitons form a sort-of "field" that gives everything gravity (like Higgs field)?

In Quantum Field Theory, a more elaborate theory in the study of elementary particle interactions;
A QFT treats particles as excited states of an underlying physical field, so these are called field quanta. At every point in space (x,y,z,t) the field is an operator which will count the number of excitations

For example, quantum electrodynamics (QED) has one electron field and one photon field; quantum chromodynamics (QCD) has one field for each type of quark; 

The field is an operator at every point in space (x,y,z,t) which will count the number of excitations, for example the number of electrons, at that point in space.
In this framework there would be an analogous underlying graviton field, and at present this has the same problems with divergences as the simple quantum mechanical expansions with Feynman diagrams.  
String theory aims to unite all forces in one viable model, but it is still at the research stage. Once this happens the analogy to the other forces will be legitimate.
A: I think it is best to answer the question "What are gravitons?" to find out what they do.  
In quantum field theory, one constructs fields from representations of the Poincare group.  The Poincare group has a rotation subgroup, so the fields have certain transformation properties under rotations, which we refer loosely to as the particle's spin.  From this viewpoint, the graviton is the unique particle that has zero mass and spin 2.  
Any interacting massless spin 2 particle must be described by general relativity.  One finds that there are interaction terms in the theory that are nonrenormalizable.  Contrary to statements that one often hears, this does not lead to any issue in the quantum theory.  One can renormalize the theory at any desired accuracy in the context of low energy Effective Field Theory, and the resulting theory is good enough to describe all gravitational processes that occur at energies less than the Planck scale.  
Thus, in the low energy theory, one can treat gravitons just like other force-carrying particles, up to differences that just come from the fact that gravitons have different spins and couple in a specific way to matter.  As with other force-carrying particles, it becomes useful to distinguish between "real" (or "on-shell") and "virtual" (or "off-shell") gravitons.  
An "on-shell" graviton is one that could be emitted from, e.g., an inspiralling binary pulsar.  These gravitons obey the relativistic dispersion relation $E^2 = p^2 c^2$, and can travel for long distances.  When they are re-absorbed by some other object, like a detector on Earth, they impart momentum to the detector, just as if the detector was hit by some object.  So the point is on-shell gravitons do not lead to attractive forces.  
The virtual "off-shell" gravitons lead to the attractive gravitational force felt between two objects.  These gravitons are "off-shell" in the sense $E^2\neq p^2 c^2$.  For example, in the nonrelativistic limit, the fact that two electrons can scatter by exchanging a virtual graviton leads to a $1/r^2$ potential felt between the two particles (as would be the case for the exchange of any massless particle).  
Finally, for very massive objects, one could think of them as being surrounded by a cloud of virtual gravitons (or "dressed").  This gravitaional dressing is responsible for the long range gravitational force felt between two large objects, such as the Earth and the Sun.    
A: Yes, I know gravitons are 'just a theory', but I'm wondering how they theoretically act.
To echo Anna's answer, but putting it more bluntly: they don't. And see how John Rennie mentioned Matt Strassler's article? See this line from it: "A virtual particle is not a particle at all". Electrons and protons don't throw photons at one another. Hydrogen atoms don't twinkle. Virtual photons aren't short-lived real photons popping in and out of existence like magic, and they aren't the same thing as vacuum fluctuations. Instead they're "field quanta". It's like you divvy up an electromagnetic field into little chunks, and say each is a virtual photon. Then when the electron and the proton attract one another, they exchange field, such that the resultant hydrogen atom doesn't have much of an electromagnetic field at all. Hence you can see the underlying correctness of the exchange idea. However when two hydrogen atoms attract each other gravitationally, they don't exchange field. Instead of the two fields (almost) cancelling one another, they're additive. So whole concept of virtual gravitons being exchanged just doesn't fly.   
Are they raining down on everything with mass to "push" other things towards it? Or do the gravitons form a sort-of "field" that gives everything gravity (like Higgs field)?
It's definitely not the former. As for the latter, you could divide a gravitational field into little chunks and call each a virtual graviton. But there aren't any actual little chunks there. A gravitational field isn't really made of gravitons, just as an electromagnetic field isn't really made up of photons.        
