Personally I always go with the ratio test. Normally I try and figure out before i've done the calculation if it's going to work or not!
If that one fails, I try the sandwich test. This does require a knowledge of which series converge and which series diverge in the first place!!
The sandwich test is particularly useful for situations involving cos and sin. For example:
and we all know that converges to .
Are there any series in particular that you're having trouble with?