From what I understand, the accepted answer is not completely true. You do not need a dual cam setup to run vtec. I believe your civic has a stock d series engine which is a single cam most, likely a d16 (if you have a d15 it is more complicated).
The difference is with SOHC there is a common camshaft for both intake and exhaust valves. where as with DOHC there are separate cams.
VTEC is a variable valve timing system which is engaged at higher RMPs so you get better performance when you need it and don't burn extra gas and risk wear on your engine riding around at lower RPMs.
Civics are probably the most commonly modified hondas this question has been asked a million times, luckily that means there is a good amount of information on it. The first google search I found returned this. but there are many more some write ups even have detailed step by step picture tutorials.
At a minimum you will need a d series vtec head, a new ecu, and the wiring harness. (and probably need to replace a few more things like gaskets while doing the swap)
While it is a good idea to upgrade many of the parts in your engine to help cope with the higher RPMs theoretically that isn't necessary, but you run a higher risk.
Your mechanic friend is probably just saying cool things you could do. I believe the civics from that year have the same suspension and braking setups with or without vtec, although there are tons of aftermarket options if you are ambitious and have deep pockets. Similarly with the DOHC vtec, he was probably saying you should swap a B series motor into the car which is a very popular swap done to civics look (B16, b18, b20, etc).