How do I run new water lines from the basement to the second floor through a finished wall?
Copper pipe should be self supporting for the height of a room as long as you clip it as close to the floor/ceiling as possible, but if you've got particularly high rooms or high pressure you still might get some banging.

There are various types of pipe clip you can get, some screw into the wall others nail, you just need a suitable anchor point.

1/2" (15mm) pipe should be adequate for most purposes, but I'd go with the size closest to the current pipes. Bath taps (in the UK at least) usually have larger diameter (3/4") to basin taps (1/2"), so you might want to use 3/4" (22mm) pipe if you've got a large bath.