Your first try would have been good if (1) you had said "multiply all that together" instead of "add all that together" and (2) you had continued for a few more steps, so you'd be multiplying 4 (coming from 17 mod 13) times 3 (coming from 16 mod 13) times 2 (coming from 15 mod 13) times 1 (coming from 14 mod 13) times 0 (coming from 13 times 13). And then you can stop, because now your product is zero and will remain zero no matter what else you multiply it by. (Sometimes, multiplying a lot of numbers is actually easier than adding them.)
Your second try seems to be based on a plausible idea, namely that 17 is congruent to 4 mod 13, so you expect 17! to be congruent to 4!. Unfortunately, this expectation is not correct.
Furthermore, the failure of this expectation can be very instructive (maybe not for you, if you already know the following, but then for other readers). Students often get bored with theorems like "If $a\equiv b\pmod n$ and $c\equiv d\pmod n$ then $a+c\equiv b+d\pmod n$" especially if the next theorem says the same thing with subtraction in place of addition (i.e., $a-c\equiv b-d\pmod n$), and the theorem after that says the same thing for multiplication. Why can't we just make a general rule that $\equiv\pmod n$ works like equality, so, just as we can replace equals by equals and get equal results, we should be able to replace "congruents" by "congruents" and get congruent results? Well, the theorems say this works fine as long as all you're doing is adding, subtracting, and multiplying. But, as you saw in this problem, it does not work when you do other things to your numbers, like taking factorials. So there really is some value in those boring theorems --- they tell you in which contexts congruence behaves like equality.