Do you know what you what to show for (1)?
presumably the operations + and * are matrix addition and matrix multiplication. given that you have probably seen 2x2 matrices before, you may be able to assume closure as given (ask your professor?).
so, your intermediate goals are these:
(a) show M2(R) is an abelian group under matrix addition. what is the additive identity, and for a matrix A, what is -A? these questions will be easier for you, if you've had some linear algebra.
(b) prove the distributive laws. this will get a bit intricate, but it's just calculation, nothing fancy, here.
for (2), you want to either prove the following, or find a counter-example:
AB = BA for all 2x2 matrices A,B
if A≠0, and AB = 0, B = 0
for every A, there is B so that AB = 1 (the identity, if there is one, not the number 1).
before launching on an attempt at proof, i advise you to play around with some simple 2x2 matrices (try ones made with only 1's and 0's).