So after taking calculus II, or maybe a first course in analysis, everyone learns a few series tests. They learn
1) Divergence Tests
2) Integral Test (from which we deduce things like $p$-series.
3) Comparison Tests
4) Root and Ratio Tests
As well as a few useful theorems pertaining to geometric series, telescoping series, a series-analog to the squeeze theorem. Maybe a few others, but that's all I can come up with off the top of my head for an introductory course. But what other series tests are out there?
I know that there are two tests which are associated, one by Raabe and one by Gauss. Another one called Dirichlet's Test. I get the feeling there are more than just these. Can anyone throw out some others for series of a single variable?
Here are some references. The bibliographies in these will lead you to many more references. Based on your comments, Bonar/Khoury's book is probably the best fit for you right now.
Bonar/Khoury, Real Infinite Series (2006)
Bromwich, An Introduction to the Theory of Infinite Series (1908) [a 1925 2nd edition also exists]
Hirschman, Infinite Series (1962/2014)
Knopp, Theory and Application of Infinite Series (1954)
Knopp, Infinite Sequences and Series (1956)
Osgood, Introduction to Infinite Series (1897)
Rainville, Infinite Series (1967)