I think you can divide the actual transport in Thailand into two categories, being long and short distance.
For short distance trips, you might take a taxi, tuk tuk or songthaew (or however to write it, meaning pick up with seats in the back). In those you will not have a car seat. But since you are only in it for a short period, it might not be worth it to take it with you. In a songthaew and tuk tuk, you will not be able to place a car seat any way.
In Bangkok you might also use the sky train. No use for a car seat in that as well.
For longer distances you will probably take either bus, train or plane. In all of these you do not need a car seat.
On the occasion you go for a guided trip. Do as mentioned below and select your driver carefully. Sometimes you don't know what driver you will get. Just use common sense. If you see the guy is almost falling asleep, using his mobile all the time or smoking. Tell him to stop with it or get out of the car and look for other transport.
Thailand is very kid friendly and you shouldn't have any problem at all in traveling.
I also agree with the suggestion of a back pack. We also have the ergo carrier and I can also recommend it.
Enjoy your travels!