Bangkok is awesome for eating, shopping and partying. Tough to beat it for a value for money city.
If you want a nice hotel to stay, I recommend Siam @ Siam. Very comfy, nice pool and very relaxing. Only drawback was the lift was suuuuuuper slow which is a bit annoying during peak times.
If you want something cheap and budget, I've stayed at Imperial Tara which is in the Sukhumvit area which Ash has mentioned is the biggest touristy and party areas. It's very close to the MRT station too so if you want to catch public transport, it's a cinch. Taxis and scooter taxis are super cheap too so it's up to you which you want to use. Traffic in Bangkok sucks the cack so sometimes the MRT is actually faster.
What you want to do in Phuket will dictate where you stay. If you want to go out at night and party till things close, you will stay in or close to Patong. I've stayed at Holiday Inn but had a Busakorn Wing room as it has it's own private pool away from general population. It's about a 10 min walk to Banglor Rd so it's just far away enough that you don't have too many drunks making noise in the area. You may also want to consider Courtyard Marriot - Patong Beach which is across the road from the Holiday Inn. I've also stayed at Sheraton Phuket but that's a good 30 min drive from Patong because on those trips, I went with family and wanted a total rancho relaxo experience.
Renting scooters/motorbikes/cars etc are at your own risk. They make up their own rules there and if you do hire a vehicle, check it front to back, top to bottom and take photos of any existing damage, scratches, dents etc before you take it away because I guarantee if you bring it back and they see a "new" scratch on it, you WILL pay for it one way or the other. Contracts mean nothing over there and your repair bill will be whatever their mate wants to charge you. If you think you can call the cops coz you're being ripped off, think twice. You're the foreigner there. How do I know this? First hand experience.