Palm Beach is an old money enclave, and is generally pretty dead, other than high-end shopping and wood-paneled dining. Worth Avenue is the place for that. West Palm Beach is a nondescript and somewhat run down small city, with nothing really in the way of attractions.

In Palm Beach County, try going to downtown Delray Beach, a bit south from Palm Beach on I95, which is much livelier than Palm Beach.

Ft. Lauderdale and Miami are easily within a two-hour driving distance (other than during rush hour) south of Palm Beach -- there is an immense amount of information on these very popular destinations on any web travel site (or FT). You can also try checking out some of the "wilder" (meaning, unspoiled, not party-hardy) beaches north of Palm Beach, e.g. the Jupiter area.

It would be easier giving you tips if you said more about what interests you.