In phuket i’d say go to Patong, koh phi phi if you can too. They are kinda party places but you can find some temples and stuff in phuket, it’s quieter than bangkok. I wasnt here for long to be honest. I stayed in balcony hostel and the staff were great there really helpful.
Bangkok has temples you can see but just a few days is enough as its more expensive and so busy. Watch for scams in bangkok and read up on them. You can take the local boat from taksin and get a day ticket which takes you to different temples. Never believe tuk tuk drivers who say they will give you a ride.for 30 baht or if someone tells you a temple is closed for a holiday, its a scam and dont get the longboat for 900 baht. Watch your bag as people on motorbikes are known to snatch them. Just keep your wits about you, bangkok is t iring. Also silom is the gay district if you Want to check that out. HQ hostel is pretty good there decent price and central for the gay district. Bangkoks mostly a party place.
Chiang mai is great a lot more relaxed, they have temples and treks you can do or elephant sancuaries. You can also go to chaing rai and pai from there. Always best to ask hostel staff or travellers you meet for suggestions. I stayed in thailand wow in chiangmai and kai was really helpful.
Hope this helps, i done more island hoping. I think the best thing is to use trip advisor for hostels and stuff. Thai people are generally really friendly you get the odd ass like anywhere. Just enjoy your travels and if i can help anymore give me a shout. =) happy travels!