I wouldn't bother with Khao San now, the government are trying to tidy the area up now and most of the market has gone apparently, i was watching a recent youtube video a few weeks ago an it seemed so different from how it was when i was there last year!
https://www.lonelyplanet.com/news/2018/08/13/khao-san-road-cleans-up/
There are still some great bars down there though. A lot of places have already been mentioned, Lumphini park is great with roaming monitor lizards and other wildlife. If you're there on a weekend day, you must do Chatuchak weekend market, if not, Patpong night market is eye opening, as is Soy cowboy, a lot of that area and Nana plaza is a decent night out too. Then there's Chinatown (Yaraowat Rd), there's far too much to fit into a full day, i've done 2 one week stints in BKK and haven't even scratched the surface yet!
Many wouldn't agree with me, but Bangkok has probably been my favourite city to visit and i still want to go back a few times just to see more of it! Hopefully in a few years, my business will be worth a bit and i can sell it and spend a year or so in SE Asia to explore more of it!