Just to add to my previous comment, if you're starting out, I would say go with thin stuff first, around 0.5 - 0.6mm
When I first started soldering, the kits I bought came with 1.0mm solder which I found was bad for learning with as it was very easy to add too much. I bought a roll of 0.56mm quickly and liked it much better.
I have 3 different sizes here: .015", .024", .031". The .031 is 60/40 and the other 2 are 63/37 (Multicore C400). I like the 63/37 better. The .015" will be used for small SMD parts (if my Gertboard ever arrives) - it's too small for some of the larger through-hole components I've tried soldering because I usually end up having to pull away before I burn my fingers on the iron and I struggle trying to control the thin wire if my hand is too far away.
Soldering isn't too bad, but I really suck if I try de-soldering a mistake. I just can't get the copper braid/wick to work on smaller through-hole components - like the ICSP header I just put on the Proto Shield and the vacuum pump (pen type) doesn't get enough to release the component. Maybe I'll have to get a squeeze ball to see if that works any better.