General Purpose Solder for Beginner's Toolbox?
For small components, pretty much any of the standard sizes will work.
In general, you should size the solder with the components. The smaller the components, the smaller the radius of solder (especially if doing SMD use very thin solder). For cases where you're soldering to large amounts of metal, it's best to use larger solder and more powerful soldering iron or it will take too long.

A general recommendation is Kester 0.010" solder for SMD, and Kester 0.025" for through hole.