The road on which Tiger Muay Thai, Phuket Top Team and Dragon Muay Thai are located is famous as shit and any tuk tuk will be able to take you there. I recommend you simply try them all out one day at a time and choose the one you like. I trained at Dragon and very briefly at Tiger.
Sinbi is a good gym, being far away from the tourist crowds is a good thing in Phuket.
The gym in Patong is, from what I've heard, a good one. But never tried it or knew anyone that did.
I personally enjoyed the less intense, but far more technical training at Dragon Muay Thai. With less students there was a high trainer/student ratio and I thoroughly enjoyed the fuckin chiiiiiiiiiillllllll attitudes. Seriously, my trainer didn't stress about anything. If you're motivated these guys will keep up with you. If you want to stop, he will stop. They feed off your hunger to train and that's how you get the best success at that gym. Cough up for private lessons and get stuck in at the group classes. Place may have changed since I went there but it is a little grimy. P.S. don't underestimate the trainers. Some have bellies. Some are like 150lbs carrying groceries. All of them are stone cold killers when the time to fight arrives.
Tiger! Bloody fancy facility, world renowned trainers and a very cool culture surrounding the place. You will make friends, have fun and probably get fit as shit if you aren't a total shitbag. They have a slick program that is not confusing to newcomers and plenty English speaking trainers. Here I would absolutely insist that you get one on ones or up your ganme and get into the intermediate or higher skill classes. Otherwise you will be lost in sea of fat tourists and two-left-feet falang.
Sinbi. Good gym as I understand. Lots of foreign fighters pass through there.
As for BJJ in Thauiland... my two cents? Don't do BJJ there. Go to Thailand for Muay Thai. If you want to have an exotic holiday and learn uality BJJ go to Brazil. Otherwise just find a good gym in your area or the US.