I know this has been asked before and I did a search. Apparently you need to open the console and

player.setrace bretonrace
showracemenu

Even after you do that though, there are problems. Is there a surefire way to change my race and keep my stats, removing my old racial bonus without my head going invisible?

Yes. You have to look up the console codes for the old and new racial bonuses. Remove the old with "player.removespell [ID]" and add the new with "player.addspell [ID]".

Not sure about the invisible head thing, as I've never changed races, but had my racial bonuses disappear when tweaking other appearance things using showracemenu and added it back successfully. I suppose you could use this to change your race without changing your race...

Hmmm. that's interesting Vermilion Sparrow. I've wondered about this too. But according to this site by PC Games, it doesn't seem like you can:

Here's what it says about the cheat the OP used:

showracemenu
Unhappy with your blind, inbred-looking Bosmer chap? You can adjust the way your character looks just like you did at the start of Skyrim – but this will reset your level and skills.

http://www.pcgamer.com/2011/11/16/skyrim-console-commands-let-you-cheat-and-do-other-stuff/

It's never reset my level, perks, or skills, but a couple of times it did redistribute my magicka/health/stamina to be equal, instead of the distribution I had chosen when I leveled up. That's also easily fixed with the console, though.

But, like I said, I never used showracemenu to change race, only to change other things about my character's appearance. I've read that what things get reset can vary based on what things you change, and that sometimes you can even get your hands re-bound by using showracemenu, so... save first?

Is there no correct way to do this? Is it really a gamble?

1. Save first.
2. showracemenu
3. Profit

You will need to console some stuff to repair afterwards, but it works.
-Loth

Use showracemenu and then fix whatever you want with the console.