The heuristics I emphasize in my AP Calculus class are as follows:
(1) Check the nth term test before you do anything else. (As noted in your list)
(2) Check to make sure you don't have a straightforward p-series, alternating series, or geometric series before proceeding with the other alternatives.
(3) A limit comparison test to an appropriate p-series almost always works for quotients involving polynomials and similar expressions (by similar, I mean expressions which have fractional powers instead of just non-negative integers, etc.).
(4) The ratio test almost always works for series involving exponential expressions and factorials. (Yes, the root test is also excellent for exponentials, but I tend to emphasize the ratio test since it's more versatile. I would not advise having students jump to the root test in most other situations.)
There are a handful of oddballs for which these heuristics don't work, but these cover a vast majority of the commonly encountered problems.