OK, let's assume you can find a beater with a blown motor for around $500.
If you're going to do any road racing, you'll have to tackle suspension and brakes, which is an easy $1500. Any 15 year old beater will likely need most if not all the bushings replaced before being track-worthy. If it's an Si shell, you should be OK with just pads, if it's a cx/dx/vx shell, you'll probably want to swap front and rear brakes, in addition to pads.
So that leaves you with ~$2000 for under the hood. If you're lucky you might be able to find a beat-on B series swap for that much. More than likely, you'll have to do d16z6 + turbo in that price range.
This is also not taking into consideration wheels and tires, which is at least another $500.
In your shoes I would try to ring some more money out of your pops
h: You can save some dough for the motor by sticking to drag racing, and skipping brakes and some suspension stuff, but that would be boring and lame.
h:
Another thing to keep in mind: it's unlikely you'll find a race series where a cobbled-together car like this will fit into an existing class, and you definitely don't have the budget for wheel-to-wheel racing. I don't know what your dad has in mind for road racing, but it would more likely end up just being track days or time trials.