I did my 95 DX, 95 LS swap and a 94 EX, 95 LS auto swap.
Both were fairly easy. If you have any mechanical skills you can do it too!
I could write a book on the process. This will help. Stick w/ 5 speed manual!
BASICS:
Integra: all mounts, axles, throtle cable, ecu w/ OBD2 to OBD1wiring harness,
tranny (GSR is best or at least 99 Si) but LS will work fine. Radiator hoses
Civic: Radiator
Tips: If you want A/C get a mount of a DOHC Del Sol. Cut a small portion of upper and
lower Radiator hose off of the civic and insert inside of integra hose for a tight fit to the radiator
This is off the top of my head, I may have missed some of the basics. Good Luck!