I'm going to Bangkok for work and am considering taking a few extra days to actually explore the city. I was also looking into renting a car and checking out surrounding cities if there's anything reasonably close that's worth the drive. Any suggestions?
There's TONS to see. You can drive up north to visit the historic city of Ayutthaya. There are lots and lots of temples and ruins up there. You can also drive south in either direction along the coast. South-west will take you to Cha-Am, and south-east will take you to Pattaya, and of course, plenty of stuff to do on the way and see! For example, there are lots of wild monkeys in the street of Chon Buri, which you'll see on the way to Pattaya. There are also lots of gorgeous beach towns along the way to those two really famous beach towns that I mentioned. I wrote more about beaches near Bangkok here if beaches are your thing: http://www.trippy.com/question/Are-there-any-beaches-in-Bangkok
I don't know how familiar you are with Thailand, but just a word of warning -- if you're not familiar with driving there, it can be dangerous and confusing, so I would recommend taking a bus or perhaps hiring a driver for the day. Pedestrians, drivers, and bikers don't always follow the rules and when caught by a policeman, the consequences aren't very scary as policemen there can be bribed with very little money, so you can imagine how messy things can get. If after a few cab rides, you feel like you can handle things, then you can think about it. Another thing to think about, though, is when you get to the smaller towns outside of Bangkok, a lot of them don't have English signs, so you would have to study a map very well before embarking on your adventure.