Ten days isn't a long time! Starting in Phuket when you really want to see other places puts an additional limit on you. Phuket is way down almost at the bottom of Thailand. A lot of the stuff that makes Thailand unique is nearer the top and northeast.
If you are arriving late at night it is not a bad thing to have at least one night booked ahead of time. Yeah, you can find some places that are Ten pounds, what's that around 500 or 600 baht? Well that is no problem in Pattaya but in Phuket a little harder sometimes. Will you settle for rooms without air and fan cooling only? For 10 days maybe Bangkok, Chiang Mai and I guess you already have Phuket booked. Some travelers need more than 10 days just for Bangkok!
Here are some travel forums that can help you find cheap 500-600 baht hotels. Make sure you mention that price range. Thailand is a lot bigger than many tourists think. Just to get from Bangkok to Phuket will take all day by bus. Overland transportation from Bangkok to Chiang Mai will take almost all day too! With your short time and desire to see as much as you can you will probably have to pay for a flight or two between the cities.
http://thorntree.lonelyplanet.com/
http://forum.virtualtourist.com/forum-19…
http://www.tripadvisor.com/ShowForum-g29…
Good luck.