It's not bad to say "I need to go to the toilet." Most natural and polite is just "I need to use the restroom." "Go for a pee" really isn't natural. When talking to good friends, "I need to take a piss" is normally what might be said. You'd never say this to someone you don't know well.

It's more polite and common to say "I need to use the restroom" or something along those lines. Generally speaking you would say you need to use the bathroom or restroom. Just talking with friends, you could say "Man, I need to pee", or even more slang would be "I need to take a piss" but that's not polite and shouldn't be said in proper company.

In a formal situation, you would just ask, 'Would you excuse me, please?", without saying where you are going, even though everybody knows. If you don't know where the restroom is, then simply ask "Where is the restroom?" They'll know what you need it for without saying. I very seldom hear "I have to pee.", even among friends. It would only be used if it were practically an emergency that you go. Usually it would be "I have to go." If you're feeling a bit more informal, the normal expression would be "I have to whiz" or "I have to take a whiz." or "I have to take a tinkle."

I think, you can say a polite sentence to go to a toilet with to say:"May I wash my hand?" You can say:"I need to go for a pee"? only to your friends or someone who close to you.