Shame you weren't here a week earlier I would have shown you around a bit, but we are heading off today back to the UK today.
Just realised you are out in the start of cool season. Lovely time to be there
Bangkok - just do the tourist thing if you only have a few days, easiest. Kings Palace, one of the large temples, floating market, look at dirty backpackers on Kao San Road.
But you can do that in a day if you time it right.
Other options are to pop over to Kanchanaburi for the day and see the bridge over the river kwai.
Lumpini Park is nice for a relaxing afternoon/early evening - don't get off at Lumpini Station, get of at I think Saphan Kwai
Go to Siam Paragon the shopping centre and see how the other half live, fascinating place.
If you want nice Thai food in a a/c place try the Central shopping malls, or any of the larger department store style shops, they generally have nice food courts with good food=, bit more expensive than the street, but if you are nervous about street food they hit the spot.
Travel wise in Bangkok - Skytrain and MRT is ok gets you around. I prefer taxis these days though if I'm off the MRT route/ Skytrain route, not particularly expensive, but can be slow depending on weather and time of day.
I used to travel by bus a lot when I was really needing to get around, but with a baby, never fancied it this time.
It's really hot and wet just now. So I imagine that the jetlag and the heat/humidty is going to knacker you. Taxis are great when this happens.
If you are up in CHiang Mai and want to explore, hire a car, it might look scary, but you just need to remember that you are going to go where you want to go and push your way in. Always look both ways at a junction or a motorbike might hit you from the wrong direction.
In theory the SRT airport train opens up on Tuesday to take you into central bangkok from the airport, whether or not it does or not is another matter.
Enjoy your trip - if you get stuck or in bother , my email is in my profile.
It's bloody expensive now there as well.