I think I want to change from an Orc to a Nord, but I am already level 24, and I do not want to start over. Can I use the console to change my race and appearance, and if so, can I do so without causing problems with my game?

You can change your race, but I'm not too sure about the problems/havoc that may cause with your game.

I think there may be minor glitches, but nothing too dramatic I would think.

The console command is...

showracemenu

However, it will reset your skill bars to zero.

EDIT :

Alternatively, you can follow this. You are essentially changing the race of the character prior to going into the race menu and avoid the bad stat resettings.

I believe you can do: player.setrace nordrace

And then you can go to the face sculptor to change your appearance.