You can really cover alot of ground in one day. Get up early, take a trip up the Chao Praya river. You can get a long tail tour boat ride for a few hours where you can see some river life on the Thonburi side. I believe you can arrange your tour at the last skytrain stop at Saphan Taksin station. Definately check out the Grand Palace, Wat Po, Wat Arun, etc. The skytrain and underground can get you from place to place easily but a tuk tuk ride is in order (just make it a short ride) and don't fall for the "Grand Palace is closed today" let me take you to tailor shop scam! For shopping, I like MBK the best but there are many, many places to shop in Bangkok. Check out Siam Paragon, Siam Square and Panthip Plaza (to see a real electronics shopping center). You can also eat at one of the restaurants on the top floor of many shopping centers. Buy your coupons first then choose your meal items from one of the many eateries. You didn't say if you would be there on a weekend but if you are then you can check out the Chatuchak weekend market. Easy to get to via the skytrain. If you are not exhausted by now you can catch a Muay Thai fight or head over to Saxophone pub to wind down with some live jazz.

Anyway, have a blast, don't get in a hurry and keep your cool (always). Also, do some research as to where things are and how to get there. I find Bangkok is fairly easy to get around and easy to do independently. All of the places I've mentioned are also easily found.