Hi. Jorrell is right on all counts. Let me share some thoughts.

Re WEIGHT:
Don't worry so much about the weight, as the car is actually lighter than it looks -- only about 3,000 lbs which isn't much for a car of its size. Much of the credit is due to the "unibody" design, where the inner panels (not the fenders, though) provide the torsional rigidity instead of the old-fashioned, heavy frame of, e.g., older domestics.
Besides, unless you do a lot of stop-and-go city traffic, the lack of aerodynamics and other factors (friction in the drive train, disc brakes, etc.)will probably take as big or a bigger toll on your "rolling" energy needs. You can minimize this aerodynamically with skirts and other body modifications, by using low weight synthetics in your rear axle and front hub grease, as well as with tall aspect ratio, narrow tread width tires, those that are made with hard rubber (i.e., promising low wear rate and poor traction, such as an H or even better, an S speed rating), and of course, a high max. inflation pressure so you can pump it up to a high pressure.

Re TRANSMISSION:
This is the one place where Jorrell (rather than made a mistake) simply may have inadvertently implied something that can be misconstrued. When he mentioned the transmission needed to keep the electric motor at its optimum rpm, actually, the highest torque output of an electric motor is at standstill -- torque diminishes rapidly as the electric motor turns faster -- so it might be thought that you're supposed to keep the motor turning fast. But there is a tradeoff, that while higher speeds reduces torque, a faster turning motor reduces amperage (energy consumption) -- so you want a compromise rpm which is generally much, much slower than a gas, or even a diesel, engine.
In many electric cars, a transmission is not even necessary unless the car is expected to cover a wide range of vehicle speeds -- from city traffic to highway speeds -- the latter being something that I expect your home-brewed vehicle will not have a chance in hell of doing unless all you need to do is drive on a highway between two adjacent exits. Don't take offense, but if you can make it work at in-town speeds, you'll have accomplished a lot! Honestly.
So I would keep it simple and start off without a transmission.

ENERGY SOURCE:
Of course, although Jorrell was aghast at the thought of you using lead-acid batteries, certain other kinds that are much more efficient will cost you a fortune, if you can even get them at any price. Other alternatives to lead acid aren't that expensive, though, and are available at a reasonable price, such as gell cell, glass mat, etc., some under the brands commonly seen in car and boating stores, but they aren't really that(!) much better than lead acid. Sure, they hold slightly more power, charge a little faster, are a little safer (maybe a lot safer) in a crash, but they aren't the(!) answer that car companies are waiting for! They're still waiting! Don't get your hopes up in that regard.
Bottom line: there are safer alternatives to lead acid, but they're a little more costly and you should not expect miracles from them.
In any case, a big factor in their long-term cost (re eventual replacement) will be dependent on how you maintain them -- don't treat them like a typical automobile battery (i.e., install and forget it). They'll need t.l.c. regarding their recharging -- you'll have to go with an expensive, multi-bank charger, the kind used on big sailboats for multiple battery banks -- and if you want your batteries to last, these chargers cost a mint! Believe me! I know!
And of course, your total stored power (range) will be limited by the humungous weight of any battery type you use, and the ability of the car's suspension to support that weight.

You may find that the distance you can run will disappoint you (speaking as someone who sees this himself on his electric-powered boat).

The MOTOR:
This is probably the easiest part of the project.
There are various kinds of motors available -- you probably could get a nice one from a shop that rebuilds industrial motors (e.g., elevators). But research the kind you want because there are many choices: DC, AC, and variants of these. They all have different efficiencies in this application.
And it isn't only their output torque that's important, as they all "spin", but it will affect what kinds of controller they'll require, as well as ancillary devices such as an inverter (if you go AC) to add to your costs.
And don't forget they're voltage needs -- high voltage (and with that higher torque) will require more batteries, which will be more costly and heavier.

The CONTROLLER:
How you're planning to control the motor will make probably the biggest difference in how efficiently your car uses its battery power. Case in point: I've got a small boat that I use on a local, small lake (more like a pond) -- I drive it with an electric trolling motor (it's quiet, cheap to operate, etc.), although small (~ 1 hp) -- well, I said the boat was small, didn't I? Sure, I don't have to worry about battery weight, because the hull carries it thanks to buoyancy. I used to have one motor that used an ordinary motor controller -- it regulated power (and the boat's speed) with a variety of resistors, much like the ventilation fan in our own Volvos. Well, with my batteries, 4 deep discharge 6 volt batteries to make a double bank of 12 volts, which weighed, by the way, about 160 lbs total, I could run at about 3 knots for about 3 or 4 hours, maximum; and at slower speeds, there wasn't much improvement, because the resistors wasted what the motor didn't. However, a couple of years ago, I switched to another trolling motor with a different controller, one that regulated, instead of by resistors, by "chopping" the current rapidly. This, I found, almost tripled my running time at slower speeds (around 2 knots). In fact, the slower I ran, the more savings I found -- at maximum speed, there was no savings (because there was no interruption). This is analogous to many modern "trolling motors", by the way.

However, for a car, we're talking about controlling a lot(!) more current, and voltage, than more a mere 12, <1 hp, boat motor. It definitely won't be easy, or at least won't be cheap, to find such an efficient controller for a car's power needs.

You can do almost anything if you have enough money to throw at it (kind of like my bigger boats :-), but do you have enough?

So, before you start this project, you really ought to look into all these (and more) factors first. Do your research, and see if you'll be able to meet your objectives.

Regards,