I have a three speed ceiling fan, but now only one speed works, the fastest. How do I begin to repair?

Look for loose/broken wires on Speed and direction switch first, bulging capacitor.
May need a capacitor or a switch.

Remote and pull chains of wall switch?

I second, is it a remote fan?

Probably a capacitor issue.

Ceiling fan capactors - troubleshooting and resources - Ceiling Fans N More (http://www.ceiling-fans-n-more.com/ceiling-fan-capacitors.php)