I've been to Thailand multiple times, but admittedly, the last time was pre covid. There are several airlines which fly internally, so easy enough to book your own flights. I can understand as 1st time visitors you might want the support of a travel agent or tour though. Booking hotels and flights from there used to be incredibly easy, with local travel agents on every corner. Also day trips etc are easy to organise there.
If you weren't going in Feb, I'd have suggested Bangkok, Chiang Mai and an island. Feb is the start of the burning season in the north so best avoided due to the smoke and smog. Anything specific you want to do? Do you snorkel/scuba dive, trek, foodies, rock climb, sun bake each day? I don't know anything about booked tours there I'm afraid. You have enough time to spend a decent time in each place- rather than packing and moving every few days. If you were into it, there are eco elephant sanctuaries, waterfalls, cooking classes etc which would be day trips but staying a the same base hotel.
Depending on your interests, you could fly from Bangkok to Koh Samui. Few days there, then the ferry to Koh Phangan. Its well know for their full moon parties, but there are much quieter parts of the island which are lovely.
Another option would be Bangkok to Krabi. Few days there, then ferry to Koh Phi Phi. I was last there pre the tsunami, so I'm sure alot has changed. From Phi Phi, you could return to Krabi, or get the ferry across the bay to Phuket. If you do go to Phuket, do not stay in Patong! There are much nicer quieter places down the coast. Kata, Karon or further south. Phuket does have an international airport, so its possible you could fly back from there, rather than going via Bangkok. I'm sure others will have better suggestions if they've been more recently than me.