Taking 60 is certainly the most DIRECT route and shortest mileage wise to get to Clearwater Beach.. But during peak season I prefer a less direct route which tends to have less traffic but is technically a further drive, it usually takes less time than sitting in stand still traffic over the bridge though.
Take SR 60 to US 19-turn left (south).. Take US19 to the East Bay exit-turn right (west) onto East Bay and follow that all the way, it takes you to a bridge (Belleair Beach Cswy) which will hit Gulf Blvd. Turning Right on Gulf Blvd will take you to Clearwater Beach, turning left will take you in the direction of St Pete Beach, it can be a nice drive to explore a little as well. Going this way also avoids the dreaded round-about that nobody seems to be able to drive.
I personally prefer Clearwater Beach over the other local beaches because of the amount of things to do, shops to visit and places to eat.