Can we always establish whether an infinite series converges or diverges? I'm currently working with infinite series for my calculus class, and I'm wondering whether we always (in theory) can establish whether a series is divergent or convergent? Of course, it might be computationally hard, but is there a class of series where we simply lack the tools to determine whether the series converges or diverges?

No, there are some criteria (for example $|a_n|/|a_{n+1}| \rightarrow l<1$) you can sometimes use, but even when you have one of those, on the limit cases (for example if $|a_n|/|a_{n+1}| \rightarrow 1$, for this last criterium), you'll have to prove it "by hand" (meaning there is no general way to do so)...
