Hello there Tony
I have travelled and stayed in Thailand for longer periods of time but I dont claim to know the whole of Thailand very well. But there are some spots I do know better than others and despite my lack of english skills I will try to give you some hints and tips - if you dont mind.
As you probably know - or will find out - Thailand is divided up into 4 larger parts - sometimes even 5.
1. Central Thailand - From Bangkok north to Nakhon Sawan, east to the Korat plateu, southeast to Trat and to the Cambodian border (sometimes the eastern parts are rekognised as the 5. part), south to Chumpon and west to Kanchanaburi and Sanklaburi and the Burmese border.
2. Northen Thailand - from Kampeng Phet in south including the northwestern provinces bordering Burma - all up to Chiang Rai - the northenmost province of Thailand and east to the provinces of Petchabun and Loei.
(this is my my favourite part of Thailand as it has a wide range of different things all from a huge variety of etnichal tribes, a lot historical and cultural influence from Burma and the old kingdom of Lanna (and a little from Laos and China as well)
A landscape that is stunningly beautiful with mountains and ridges, jungle and caves, rivers and waterfalls and hot springs. Temples both more modern and those who are many hundred years old.
3. The northeastern part of Thailand - also called Isaan. From Korat (Nakhon Ratchasima)in the west north to Loei, east to Nong Khai and Nakhon Phanom, south to Ubon Ratchatani. Big parts of Isaan is bordered from Laos by the mighty Mekong river. Most population here are more or less of laotian heritage and speak a dialect close to laotian and their culture (food, music etc etc) are somewhat "laotian". An exception is those in in the southern provinces of Buriram, Surin and Si Sakhet - where you can find lots of people speaking cambodian language. In these provinces you will find old khmer (cambodian) temples/temple ruins, especially the Kha Phanom Rung in Buriram province (restored in the 60s) and the Kha Prasat Wihan(ruins)in Si Sakhet province.
Its and interesting part of Thailand as well but the landscape are not as spectacular as in northern Thailand.
4. The southern part - from Chumpon and south to the malaysian border.
I am sorry but I dont know this part of Thailand very well....
About car rental I am sorry I dont know a lot. I have been travelling mostly by motorbike or bus. But there sure are plenty of possibilities and I think that it will not be a big problem for you to find a decent car for rent.
If I was to suggest a route to you - lets say from Bangkok and to the north and northeast it could be like this:
First a little comment about that what you say you dont wont like to go the "touristy stuff". Well if there isnt anything to watch or experience, there wont be any tourists there. On the opposite - if there is anything of interest in a place there will always be tourists there. You cant get away from that fact.
Dont be so afraid of going places other have been, there will be a lot of sights and experiences left for you still.....
My suggestion is first to go to Kanchanaburi, a town just around 3 hours drive west of Bangkok. Its a detour from the north but could be worth spending a couple of days. The "Riwer Kwai bridge" is here, the "death railway" and some very well maintained war cemeteries. Lots of history from the second world war. Further you can take trip up to Sanklaburi and to the "3 pagodas pass". Its not such a interesting spot itself but the tour up there is nice. The Erawan 8- layered waterfall is on the way up there.
Kancanaburi is a nice place to soak up the spirit of a mid size provincial capital and town in Thailand.
Then back to near Bangkok you can go to Ayuthaya with a lot of old temples and temple ruins. Further from there its easy to suggest that you go asian hw 1 to Tak then turn left and so more or less follow the nearest road to the border all the way from there all around the whole of northern and northeastern Thailand all the way to Buriram and Sa Kaeo province - with of course a few detours to main hubs and attractions.
Mae Sot, Mae Sariang, detour to Doi Inthanon - Thailands tallest summit/mountain. Then to Mae Hong Son - the least populated province of Thailand. Its just a pearl and must be experienced! Then to Pai - the little "hippie" town of northern Thailand. In this area there are plentyful of sights, detours and things to do such as trekking tours, waterfalls, caves, elephant camps and riding, tribal villages etc etc.
From Pai to Fang and Doi Mae Salong where there is a huge flower plantation (thais love this place) and to Chiang Rai - the northernmost province of Thailand. The Night market in Chiang Rai is beautiful. From here you can go to Mae Sai - border town to Burma and then you can go to "The Golden Triangle" where Thailand, Laos and Burma meets up and China isnt far away. There for example is an interesting opium museum there. Chiang Kham and Chiang Saen is nearby and interesting spots as well. Phayao province/town is also near with the biggest lake in Thailand.
Then you can detour to Chiang Mai - the second biggest city in Thailand and cannot be missed. Its not at all as big as BKK and still have a somehow "smalltown feeling". Very charming indeed. More information is easily available.
Then you you can go to Nan province - maybe via Phayao again. Nan is one of the most remote provinces in the whole of Thailand and hosts mountains and jungle and the roads through here are just spectacular - belive me! Then further to Loei province that has two of the better national parks of Thailand namely Phu Reua and Phu Kradung. Lots of stuff to do there.
Further from Loei you can do the road near the border to Laos along the banks of mekong river - in fact you can follow the border and the Mekong river almost all the way "around" Isaan except from a few parts here and there. Nong Khai is the main hub to Laos and a pleasant experience. Dont miss out the hindu inspired temple of Sala Keow Khu with stone figures up to 30-40 meters tall nearby Nong Khai. Udon Thani is also a pleasant provincial capital nearby. Then Sakhon Nakhon, Nakhon Phanom with a very nice riverview area with restaurant, bars and hotels, further to Mukdahan and Ubon Ratchatani, then Si Saket, Surin and Buriram with the before mentioned khmer temples. From Buriram to Korat and visit the huge Zoo outside Korat could be a suggestion.
From there you can drive south to Sa Kaew province and all the way down south to Trat and Koh Chang. And from there up to Bangkok again.
Detours to for example Sukothai from Chiang Mai with ancient temple ruins, to other provinces such as Petchabun, Phitsanoluk is also recommended. Detours "inland" in Isaan is also recommended, Khon Kaen, Roi Et for example.
I do think that to kinda follow the border around northern and northeastern Thailand - with decent detours here and there - will be a interesting route and tour. You will be able to see and experience a whole lot of these parts of Thailand doing it this way.
But its only my humble suggestion to you and if you have any questions - just feel free to ask...
Regards from Snakeboy