Can anyone tell me how many levels of upgrade there are for the engine, transmission, aero parts, tires, etc.? And are these available all from the start?

If I remember correctly there are up to 7/8 level upgrades for certain parts. Usually it goes up to 4/5 if I remember correctly... I'm trying to remember its been close to 9 years lol.. Also no you can't purchase level 7/8 from the beginning you have to earn them and buy each previous stage before being able to purchase the next stage. Meaning you need alot of money to be able to fully upgrade the expensive cars like the S30 and the Porsche.

in Zero its 7/8 in Three its like level 9/10/11

Here are the maximum level of upgrades for each part:

Engine - 5
Muffler - 7*
Drive train - 4
Suspension - 4
Chassis - 4
Wheels - 9
Front bumper - 5*
Side skirt - 3
Light - 1
Grill - 2
Mirror - 2
Hood - 2
Rear bumper - 3
Rear spoiler - 5*

*must be unlocked by defeating certain bosses

Most FAQs will tell you not to bother with aero parts since they are more for appearance than performance. This is pretty much true since they don't have much impact on your car's handling. However, you can still attain some weight reduction and probably improve acceleration or top speed by a tiny bit, if ever.

Actually, I have found that in this game, the aero parts DO have an impact on your handling by a bit. I seriously play it every day, so I notice things like this...
Anyway, things such as cars with a big spoiler have great straightaway speed and acceleration, but do terrible in corners...while cars with no spoiler or a really little one get through corners beautifully. Not as well of a straightaway performance, but sort of close. I guess the spoilers just make the rear heavier? Haha.

sorta of reverse of real life eh?

Yeah I know! I didn't understand it exactly...I just sort of figured that the spoilers made the rear so heavy that it slid out. Like on the R32, there was this HUGE rear wing, but if you put it on, the car slid around everywhere and had no traction for a grip run, but there was also a flat wing that looked like the car didn't have a wing (like Reina's GT-R) that gave you perfect traction.