Since I don't feel the original question is answere well enough yet, here is my answer
Carsfor around 5000 or less that are good for tuning included, but aren't limited too:
89-94 Eclipse, talon, laser: Try to find an awd versio0n becdause they will be the most fun and are easily tuned.
94-97 Acura Integra: Great power plants the b18 block that comes in these cars is extremely customizable and has a great following in both the US and Japan, so parts will be plentiful.
Pontiac Sunfire/Chevy Cavalier: Great first cars, mild tuning can be done for little money, and without reliablity issues. However if you want to get into high power ranges, be prepared to sen some serious cash. These are probably one of the cheaper cars you will find.
Honda Civic: Like the integra, these cars have a huge following and are a great staring point. Not only do the stock d and z series motors have some potential, but the B series from Acura is literally a bolt in application in most cases. I would not suggest getting any of these made before 1992, mainly due to rusting.
1990-1999 Celicas: Great cars, Toyota Reliability, and good style with both generations. These cars can be made quite fast with the right parts and a little know how. The 1994+ Celicas can be outfitted with a 3sgte motor with little work, and you will not regret it if you do.
Ford Mustangs: A great bang for the buck car. You are going to have a lot of power right out of the box, especially if you opt for the V8. Parts are readily available and your mods can range from Milwd to wild with prices ranging accordingly.
Subaru Imprezas: An awesome first car, actuall this was my first, and favorite, car. I had a 1999 2.5rs, which I still have now 3 years later. these cars have the advantage of being All Wheel Drive, so handling is above average already. Also, If you are lucky and find a 2.5 like I did, they have an ej series motors. The ej is what is in the wrx and the STI, and can be transplanted without any motor mount changes. If you can't find a 2.5 in your budget, the 2.2 L version is a good alternative. Although you cant get the hug power you may want out of it, they are a good starter car, and will accept the EJ motor with some work.
Nissan 240sx: RWD, and very potent motor potential. Just like Hondas, Nissans have a huge following, and thus parts are widely available. Te ka24 motor that will most likely come in it is a good baseline. Also, if a ot of power is desired, the more potent sr20 motor is a viable swap, meaning it is do-able without much modification.
I realize this isn't every car possible, but These cars I listed are great starters, and most you can find for right around 5000 or less in some cases. Hope this helps