What is the right amount of exclamation points in a formal email? Where are they appropriate if you want to come off as friendly but professional?

I would say that normally formal communications, whether an email or on paper, would exclude exclamation points. Exclamation points are intended to convey surprise, alarm, excitement or occasionally humour, none of which are typically required in a formal email.

To convey an friendly but professional attitude, use shorter sentences and simple vocabulary, but avoid slang.

I do agree with the above answers, but I would just add that I think there are a few places where it is acceptable to use exclamation marks, such as in a spot where you are thanking someone, eg, "Thank you for responding so quickly!"
Or in a "salutation" sort of spot, such as, "Have a great day!"

I would suggest to just not use any in the main body of the email.

Just because you're being professional doesn't mean that you can't be friendly. You're far more likely to create better working relationships if there is even just a little friendliness in the email.
:)