Annie,
Check the 'accomodation' option, left panel. Thailand, then BKK, budget. There will be dozens of choices in your price range.
I think you want to 'trek' not 'trike' in Chiang Rai. Many companies offer services, I don't know which one is best. I would not book anything until I was there. Hopefully you will chat with others before you get there and perhaps they will recommend one over another.
Of course you can hit Koh PhanGan or Koh Tao instead of Samui. Since you have been to Samui, I am going to vote that you spend those days over on the Krabi coast. It is cheaper to visit Koh Phi Phi from the Krabi side than the Phuket side. Yes, you can stay on Phi Phi. I suggest you visit Maya Bay on Phi Phi Ley. The movie 'The Beach' used Maya as the 'perfect beach' Leo DiCaprio was looking for.
If you have not been to Railay Beach & Phra Nang yet, I would day trip them from Krabi town. Just beautiful. You might also consider hiring a longtail boat while at Railay and islandhop. There are dozens of uninhabited islands offshore that will be worth your time to visit. This can be one of the most memorable things you do while in Thailand. Once you have seen Railay, Phra Nang & Phi Phi, Phuket is going to be a disappointment. Expensive, crowded, not as pretty.
Good luck with your plannning.