since I've just finished the same process as Earl above (ironically on another 240SX), I'll provide some short insight as to what it cost me to build the car...
Car itself: $1400 (ebay- blown engine.)
used OEM Engine: $250
Tires: $350 (used Toyo R888)
Roll cage: $3200 (very nice 6 pt cage by local race shop)
Race seat: $300
Fire system: $400
total: $5900
That's the ABSOLUTE CHEAPEST you can build a car unless you do the cage yourself.
Since you already have the car, you can knock off $1650 from that price, but you're still looking at $4500 or so to get your car on the track in its current condition.
you will still need another set of race tires, spare wheels, better brake pads, new rotors and brake lines, probably new calipers since the car is oooold. suspension? sky's the limit, even on an IT budget. Plan at least $1000 for a decent suspension.
All said and done, I have about $12,000 into a car that still has a 100,000mile factory engine in it with a header and intake. zero internal mods. I have a ton more done to it now (suspension, brakes, sway bars, etc) but it's still no faster than a well driven ITS car.
OR... You could take $4500 and go buy the Integra linked above and run side by side with my car. You'll have a MUCH faster and better handling car with a lot more go fast parts than if you were to build it yourself with new (or even used) parts.
Keep in mind you'll still need a racing suit, helmet, shoes, gloves, etc and a driving school. Add $2000 to the tab right there.
That will get you to your first race as a novice. the bill goes up from there.
Im' not trying to discourage you or be pessimistic about things, but I'm being realistic about racing. Road racing is NOT cheap- even in a "cheap" car. A full weekend at the track will cost about $750 by the time you pay for entry fees, fuel, transportation to/from the track, food, and the most basic of wear and tear on the tire- tires and brakes. And that's assuming you don't break anything or flatspot a tire.
Hope I didn't rain on your parade, but I don't want anyone to be disillusioned by "cheap" racing. Cheap is relative to all of us. What may be cheap racing compared to a STO or GT car is absurdly expensive to the young guy that thinks he's going to bring is daily driver/autocross out and road race.