The short answer to your basic question is no; but a lot is known (and written down in various places including a 1978 paper in J. Algebra by D.J. Glover based on his A.N.U. thesis).   The long answer is that the bookkeeping involved even for this small case gets quite involved, a little more so for the finite general linear groups than for the special linear groups.   

There is a detailed survey with references in Chapter 19 of my 2006 LMS Lecture Note Series 326 *Modular Representations of Finite Groups of Lie Type* (Cambridge U. Press).   The emphasis throughout is on the "defining" characteristic $p$.   In Chapter 19 the main theme is the decomposition of all symmetric powers of the natural representation for general or special linear groups of any rank, specialized to rank 1 in 19.7.  

Even in the rank 1 case it's unreasonable to expect explicit closed formulas, but the fact that weight multiplicities are 1 already simplifies the problem a lot.  A nontrivial test case involves the multiplicity of the trivial representation in each space of homogeneous polynomials in two indeterminates: this leads to the *Dickson invariants* (for the finite groups over arbitrary finite fields).  But this already requires a lot of ingenuity.  As far as I know, the approaches described in this chapter are the only ones so far attempted.   

Concerning your added question about tensoring two symmetric powers, this must get a lot more complicated to work out in detail.   Since representations in prime characteristic generally fail to be completely reducible, you start to run into a large assortment of indecomposable modules including the projective ones.   But if you are only asking for composition factor multiplicities, there is a better chance of obtaining these (in principle) from study of the individual symmetric powers.      

It's important here to take a close look at what has already been done, since the results go back all the way to older work of Dickson and involve quite a range of methods in the modular representation theory of finite groups.