Welcome to the site.
For a start, have you tried looking at the
Ariel Atom? Wicked little car that you might get some good ideas from.
Wander around the projects in progress a bit. There are a lot of different ideas to be found in there.
Also check the suspension section of the site. Lots to learn.
You're going to have to take a lot of what you learn here and apply it to a different application idea than the majority of the machines on this site - most are off road oriented.
There have been some good on road design books mentioned recently. Search that.
I believe that there are a couple of other rides that are street oriented here. Once again, search. (yea, I hate that answer too, but in this case, it will be your friend)
Okay, so you've decided on a power plant - do you have it already or are you just a Suzi guy and want to use it?
If you already have it, what do you have? (as in what parts are with it) You'll need what's known as a "power package" (or something along that line - works when you do a flea-bay search).
You'll need pretty much everything out of a bike other than the frame, bodywork and suspension stuff. Since you want it to be street legal, the wire harness in the bike will contain everything that you'd need for running - can control the engine, lights, signals, etc.
Might be able to incorporate the stock shock, but you'll need a second one and then you'll also need another pair for up front. (I'd assume that you'd use the stocker for the rear end as per it's higher rate spring and such)
You'll need the exhaust system - or at least want it to use as patterns, maybe use the headers and such. And while I'm on the exhaust idea - make it quiet! Loud machines do nothing but piss of the people around you and bring unwanted attention. Louder is NOT faster. (unless you have Mustang with a 5" tach and side pipes, then it
has to be faster...
- never mind....)
You'll need the entire intake system - Throttle bodies, ECU, get the airbox too. It will have sensors and such in it - and since you're road going, it may be a good thing to just incorporate into your build. May as well try to keep the fuel tank off the bike too - it will have the fuel pump and plumbing in it.
What sort of budget are you playing with? That may decide a lot of what you do for running gear. Do you want reverse? That can cost a bit - RPM, Jeffco and Quaife would be the transmissions to look into. Do you want a diff? Once again $$$ You can do a closed diff that's driven by chain, but that gets rid of reverse (or at least a reverse box). Reverse can also be accomplished by flex plate and starter motor. See Yoshi's posts for this stuff.
What about the chassis? Tube frame I'm guessing?
Running gear - brakes, steering, shifting ability.
Lots of stuff is required, but look through the site and you'll see that lots of people are doing it.
Once you get an idea of where you want to go and what you want to do, start a "Projects In Progress" thread and document the build from the very beginnings up through the completion of the car. It's fun to head back and look at where you started once you've finished the thing. You'll be amazed at how much you've learned.
One other thing - go find all you can about suspension definitions. King pin angle, camber, caster, toe, ackerman, scrub radius and more are just a few things to learn about.