The must do tourist draw is the palace and the Buddha at Wat Po. Lots of tourists and crowds but very interesting from an architectural and cultural point of view. Depending on your pace, it's a 1/2 day to do both (they're close to each other). Go early! The sun will fry you otherwise.
Go to the bar at the top of the Banyan Tree hotel to catch the sunset. Go as soon as it opens for the night or don't go at all (you won't get a seat and it'll be a useless visit).
It has a dress code which is not strictly enforced but I wouldn't risk it. Close toed, non-sports shoes, pants (not jeans), collared shirt. It's up 60+ floors in the open air and it's something to see.
Best seat (in my opinion) is at the bar corner with your back to the sun or facing the sun. If you're kind/nice to the bartender, they'll make sure that you get a drink when you need one. If you're a jerk (like some folks were that I saw) the bartender will just freeze you out and you'll be waiting. They do dinners there but it was overpriced for what they offered (i.e. 3-4 star food at 5 star prices).
Check out Khao San Road if you want. It's the backpacker/drinking kind of area. It's cool if you're younger and into meeting travelers, etc. but it's gungy and I'm not really big on drunk tourists who think they're in Vegas. Cool to see but not where I'd want to be for very long.
Take the river ferry around. It's fun. Jump on, jump off. They'll try to get you on a "longtail" or a tourist boat but, in my opinion, it's way more fun to cruise around with local folks. Very pretty near sunset, obviously. You can take the river ferry close to the palace or Khao San rd. I think it's like $1 one way.
See if you can find a decent priced spa if you're into that. Or see about getting in to a hotel pool for the day - there are a couple hotels near the river with great pools and river views. I'm sure you can pay for access. Or just stroll in, blend and try to find a stray hotel towel so it looks like you belong ;)
Bangkok also has a couple regular kickboxing events that are apparently worth checking out. I never had the chance. They have them in Chiang Mai as well but apparently the ones in CM are basically sparring matches put on for the benefit of tourist money. Bangkok offers legit bouts to watch. Google for more info on that - not hard to find. Just make sure you go to the right boxing stadium location - one of them has moved from its old location which confuses people.
I dig on street food but that's up to you. My rule is: try it if there is a line up (popular means higher turnover so fresher) and it looks properly cooked. Mago sticky rice and banana roti are my two favorite meals (both also found in Chiang Mai).
Re: gear. I don't know which is the best spot at all. A complete guess would be Bangkok because it is a huge city.
Hope that helps - and really hope you enjoy your trip. I miss that country.