I know that if I want to prove a set is linearly independent, then I should prove that the equation $c_1v_1 + c_2v_2 +...+c_kv_k = 0$ implies $c_1 = c_2 = ... = c_k = 0$.
However, if I am asked to determine whether a set S is linearly independent or dependent, should I do the same but show that they mustn't all be 0's (i.e. infinitely many solutions) or should I show that I have a vector that is a linear combination of two others?
On
You have already your answer in the first sentence. Try to prove that your set is linearly independent. If you fail, then it must be dependent (I mean, if it turns out to be false, then it must be dependent). For example, are the two vectors $(1,1)^t$ and $(2,2)^T$ linearly independent or not ?
Both ways are theoretically correct, but in practice, you usually have a basis so you can write every vector $v_i$ as a column of numbers $(b_{1i},\dots b_{ni})^\top$ so that the condition to check linear independence is a single homogeneous system of linear equations $B\,c=0$.
Then you get that system solved by your favourite method (Gauss is a good candidate) and you obtain a definite answer: either the only solution is $c=0$ (thus the set is linearly independent) or there are infinite solutions (hence the set is linearly dependent).