You don't need a special visa. They will just stamp your passport like any Euro country if you are a US citizen. If you are going to the beach, you can wear shorts or bikinis or whatever. There will only be foreigners there. In other places I sometimes see foreign men with shorts, but for women it is discouraged. Otherwise dress for the weather. Dubai is quite cosmopolitan, but some of the other emirates like Sharjah are not.

I always recommend getting a car to get out and see the country. If you like golf, there are some great courses. Or you can rent a 4 wheel drive and head out to the desert. There are some good dry river beds for 4-wheeling. You can also head out to Hatta or Al Ain. Darren's recommendations are good too.