If you want power, go with the Integra. For the Civic, unless it's an SI, you can't go far on stock internals. This is what I regret the most with my Civic, I want power now, and wish I had gone with an Integra instead.
To make things easier too, go for an OBD1 Integra (94-95), so all you'd need would be a chipped OBD1 ECU (Save ~$80 for a OBD2-OBD1 conversion harness).
Three things should be considered. Weight, Traction, Tune.
Weight: Get the lighter trim available (I believe a 94-95 RS hatchback ~2500lbs full interior/accessories as opposed to ~2600-2800 for the higher trims and later years). Upon stripping the vehicle of dead weight, panels/seats/accessories, lighter wheels, etc; you can run better times.
Traction: Look into getting a B-series LSD Transmission
You want better gearing and power to both wheels instead. Tires are just as important.
Tune: Get a reliable tune. This is something that has to be looked at locally, or you can self-tune it if you have the patience and willpower to learn.
First of all, repair it so it's reliable and safe...
The whole project could run you anywhere from a low $5,000 - $9,000+ GOOD LUCK!
http://www.itrsport.com/specifications.html