Here is the easiest way to make big money and horsepower in this game. Once you start a new profile, you get 5K to get a new car. Goto the used car section and purchase the cheapest car there (like under 3K). DO NOT spend the leftover money on rims and paint, but, usually the junker you buy looks like a rust heap so if you must you can paint it the Rattlecan $100 paint job and don't get any decals (it's an extra $300 for decals). This won't make you any faster which is why I don't recommend it, but it does knock up your respect a few bars which is good. If the car you purchased comes with a radio (typically a basic AM radio) remove it and sell it along with the fog lights (neither are nessesary for your car to function or enter races). You could also sell the taillights and headlights, I guess.

Now you have the car, choose the mods you want. The modifications that yeild the greatest HP gain-per-dollar are the Race Cam, High Flow Air Filter, High Flow Fuel Rail, Platinum Plugs and Wires, and High Flow Cataylitic Converter. I recommend looking through the used parts section and see if you can find any of these parts in there. The only problem is they might be of bad quailitly or not even work at all. Thing to watch out for in the used part section would be a part that would commonly be very expensive for very cheap (I.E. a Stage 3 Supercharger [retail is $3200 new] going for under $1000).

A few other must-have mods in the beginning are the inexpensive Race Sway Bar kit and Traction Bar Kit 2 (combined cost is something like $400 or less if you buy new) and they will help you get off the starting line (holeshot) better. Rear Race Gears, Race Driveshaft, Race Clutch, and Pro Race tires are also good investments on your way to the top.
Another tip; Pro parts like the Pro Radiator and the Pro Oil Cooler don't make your car faster- these parts just last longer than the stock radiator or stock oil cooler (usually twice as long), which is good. Always check up on the oil too- it wears down fast.

The used section of the game (cars and parts) changes weekly. So if you are on the hunt for a certain part, you can wait for the part to turn up in the used section or go buy it new. If you're on a tight budget and you have to have the used part, then here's a tip to make the week go by faster. Remove your engine block and reinstall it. The block takes 24.0 hours to install and there's 5 hours in a day, so most of the time (if you start late in day 1 or day 2 of the week) it will be a week-long project. You don't lose any money or repuation or anything by doing this. Once the week passes, go back to the used part section and look for the part you need. If it isn't there, uninstall and reinstall the block again and keep doing so. It is bound to turn up in the used parts for sale sooner or later.