Basically just what Michael Lugo said: do the easiest test first, then the second easiest, the third easiest and so on. I was taught [in undergraduate physics (credit to my lecturer)] to do these three tests in the order laid out:
The ratio test for convergence (d'Alembert test):
Assume $a_{n} > 0$ for all $n$ . Let $R = \lim_{n \to \infty} \frac{a_{n+1}}{a_{n}}$.
If $R < 1$, the series $\sum_{n=n_{0}}^{\infty} a_{n}$ converges.
If $R > 1$, the series $\sum_{n=n_{0}}^{\infty} a_{n}$ does not converge.
If $R = 1$, the series $\sum_{n=n_{0}}^{\infty} a_{n}$ may or may not converge.
The root test for convergence (Cauchy test):
Assume $a_{n} > 0$ for all $n$ . Let $Q = \lim_{n \to \infty}$ $(a_{n})^{1/n}$.
If $Q < 1$, the series $\sum_{n=n_{0}}^{\infty} a_{n}$ converges.
If $Q > 1$, the series $\sum_{n=n_{0}}^{\infty} a_{n}$ does not converge.
If $Q = 1$, the series $\sum_{n=n_{0}}^{\infty} a_{n}$ may or may not converge.
The integral test for convergence:
Assume that $f(x)$ is a monotonically decreasing and positive function on the interval $N \le$ x < $\infty$, and let $a_{n} = f(n)$ for integer $n$.
If $\int_{N}^{\infty} f(x) dx$ is finite, the series $\sum_{n=n_{0}}^{\infty} a_{n}$ converges.
If $\int_{N}^{\infty} f(x) dx$ is infinite, the series $\sum_{n=n_{0}}^{\infty} a_{n}$ does not converges.
Also, sometimes if you have an easy looking sum, you can tell by inspection.
Another note: always always take the limit to infinity; testing the first $10^{10^{10}}$ terms, you haven't proved that it converges.