If you know a bit of algebraic topology Hatcher's book mentioned above is great. I thought it was a little slow going, so instead I prefer Mosher and Tangora, which is now a Dover book!
I think there are some great exercises that really made me feel comfortable with Spectral sequences. The first Two SS I would look at are Bockstein and Serre. First look at the description/construction/definition of $sq^1$ in MT. It IS the first differential in the mod two bockstein SS. so now you can compute integral cohomology from mod two cohomology once you know how $sq^1$ acts (this also shed light on $sq^1$ for me). To do these computations you should adjoin some indeterminate and use adams grading. then $d_1$ goes up one and to the left one.
Now you can use the serre SS and various simple fibre sequences to compute the cohomology of all the complex and real projective spaces WITH their ring structure! (first compute it for the infinite cases then restrict). This is all spelled out in MT (maybe not the part about restricting to get the finite dimensional projective spaces).
Then compute the cohomology of $\Omega S^n$ for every n, with ring structure!
Then prove the Thom Isomorphism Thm with a relative form of the Serre SS.
Also, when you are looking at the bockstein SS draw one of boardman's unrolled exact couples and look at what is going on. And dont use their construction of the steenrod operations, there are better places imo.