I just have to plug one of my favorite freeware programs, Hoversnap:
Linky
Just start this before you start the game and then every time you hit Prt Sc button it will automatically save the screenshot in whatever format you chose in the options. It will automatically name them however you tell it to. It may cause a little, very brief pause in the game (really only noticeable during fleet battles and perhaps if zoomed in on a rotating planet), but does not affect performance at all. I used it last night to take about 9 shots while playing and it worked perfectly.
This way you don't have to interrupt the game, open Paint or another program, mess around with pasting, saving, etc. Just hit Prt Sc and keep playing.
Edit: On that page you will have to scroll down a bit until you get to the Hoversnap link.