Bangkok is a city I absolutely love. There is a hell of a lot to do there though. What are you most interested in doing?
Going to Ayuthaya is a good shout. The trains are pretty slow, and It's much quicker by bus; but, if you're interested in a train journey and are willing to wake up early enough, it would be a good day trip.
I also very much enjoyed my visit to the snake farm. It's run by the Red Cross to make anti-venom for people who suffer from snake bites. It's worth a good hour or two as you'll learn lots about snakes.
I'd also recommend staying away from the river. Whilst I've never found taxis in Bangkok to be that bad (only once did anyone try to cheat me with a dodgy meter), the traffic can be horrendous. A hotel near the sky train would be better. My preference is somewhere around Silom or the top of Sukhumvit (Say up to Soi 17 or so). There are a few 'seedy' streets like around Patpong market or Soi Cowboy, but otherwise these are great areas with plenty to buy, see and eat with great public transport connections.