Avoid lead free stuff.
Stick with RA type flux Core such as Kester's #44.
I have gone through several rolls of Kester and have a nearly identical roll of solder from AIM. Made in Mexico - cost a lot less but smells like dog crap when it melts. The Radio Shack stuff works well, also. I still have a roll from the 1980s that works fine.
63/37 works a bit better but costs a hair more than 60/40. The 63/37 composition puts the solder at the eutectic point. This pretty much means it transitions nearly instantly from liquid state to solid state. Other compositions have higher amount of time when they're in the 'plastic' state which can result in bad solder joints if moved. 60/40 is nearly on the Eutectic point, 63/37 is right on it. Most people won't notice a difference between the two.
Like Forceflow said -- 0.031" for most PCB work, 0.062" for large items such as lugs on coils.
And like everything else, the cost of solder has gone up this past year. It is now just over $20 for a 1 lb roll.
A 1lb roll is pretty much a lifetime supply for most casual users. You can also get this stuff in much smaller pocket packs.
Best price I can find for name brand stuff is Kimco.