I'm having a lot of trouble with proofs and rules of inference, If somebody could talk me through each of these problems I will be grateful. Proofs are very interesting to me and I know I need practice, but help is always appreciated.

* = conjunction

(P > T) * P

(T v S) > (T > R)

------------------

(T*P) * R

Let's see if I can write this the "usual" way AND "translate" it. You do the formal writing: -- for this to be true both sides must be true, so P is true but then also T has to be true for to be true. -- The left side here is always true since T is, so the right side must be true as well, and since T is true also R is true --------------------------------------- -- follows at once frome the above
R > (Q > S)

P * R

T > P

T v Q

-----------------

SvP

Hint: the second line tells us that both P, R are true...and this is pretty much you need! Now you try the other ones by yourself. Tonio Ps. Of course, you must thoroughly know the truth tables for the different connectives...!
(P v Q) > (P * (S v R))

P v (T * R)

(T * R) > S

~S * (P > Q)

----------------------------

Q v S

Thank you.