I have a soft voice and want to mic myself singing when I am playing piano at home with friends. I know I need a mic and stand. What do I connect to? (Straight to small guitar amp? Some sort of pre-amp and amp set up? Little karaoke system?) I've never been in a band or anything and am clueless.
Don't get a guitar amp. Get a monitor.
Here's a monitor for vocals.
http://www.sweetwater.com/store/detail/VSM200XT/
You'll have to get the mic, mic stand, and an XLR cable to plug it into the monitor.
If you don't have a microphone yet, this was the standard for years, used on American Idol, Jay Leno, all the late night shows and used on stage by many bands for many years until about 2006, when technology got a little better with condensers.
http://www.sweetwater.com/store/detail/SM58