Here's my 2ct:
In order to create a good car, a lot of thought and testing needs to go in there.
First, you start with the body.
Get low resistance tires, preferably with light weight alloy rims. Scooter rims are good for this, but small bicycle rims are also very good.
Second, determine your lightest weight, preferably smallest in size driver. Small in size means smaller bodywork (cockpit) is possible, and that means lower in weight.
Once you determine your track (know what kind of road, weather condition, winds, and hills) you will have to face, you can start building the body to the track.
Is it windy? (more effort in aerodynamic body)
Lot of potholes and rough roads? (better suspension)
Mainly asphalt or track roads without potholes? (get no suspension, or a very simple suspension. The lighter the body, the better the MPG).
For that reason I'd suggest you guys to make the frame out of aluminum, and weld mounting points to it, and make the body plating out of plastic, instead of solid metal.
Lots of uphills? (match the engine size to the performance needed. That being said, 125cc engines are IMHO the best of both worlds; great MPG, while at the same time it has the performance needed to shoot your cars at 50MPH through the streets).
If your engine size is pre-determined, not much you can do about that; but you can get aroud some things, for instance this:
If the engine's size is limited to a certain cc's, you can actually artificially increase performance, by installing a compressor, turbo, or supercharger.
If it's a small engine, use a 12V battery powered compressor, to artificially increase PSI in the cylinder.
Higher compression, means more performance.
Choose good fuel, BP 93 octane fuel is the best out there; and good engine oil (you don't need the engine to last very long, so 5W20 is a good starter oil).
Transmission oil, if you need it, doesn't need to last very long.
Instead of 80W90 which is standard transmission oil, equip it with heavy diesel engine oil, 20W50 or something; the vehicle just needs to survive a few races, not 8k miles I presume.
Thicker oil makes your vehicle last longer, but lowers MPG.
Run the engine lean.
For this you'll need some measuring equipment, and advanced knowledge on engines. How lean can you run the engine, depends on if it's water or air cooled; what the outside temperature is, and when it rains, you can run it leaner than when it's dry.
So choose the right mixture. Jet size needs to be increased a bit, when you're using a compressor, turbo or supercharger, just not to run too lean.
Again, running lean is bad for the engine, but great MPG's can be attained, at minimal performance losses.
If you're not going for a compressor or turbo, you could just decrease the jet size.
It'll all depend on how much value is put on getting first of the race, and how much is put on great MPG.
The final gear should be determined when the car is almost finished. It should be so small, that the engine can barely increase the vehicle's speed.
In fact, the best thing would be to have an overdrive gear, that in windstill would be able to maintain certain speeds, but the gear lower should be a few teeth lower, so you can actually accelerate with it.
The final gear ratio also depends on the track. If it's mainly long stretches of continuous riding, then that. If it's a lot of small corners, and hills, you must find a final gear ratio, that will be able to bring out the best performance out of the car, while still be able to rev the engine as low as possible (sweet spot ~2-2,5k rpm for 4 strokes, 2 strokes I don't know).
Don't tune the gears the way car gears are tuned (in an audibly nice way). Instead Tune them in a different way; Plot out your engine's torque graph.
At maximum torque peak 7500 RPM (say torque peak is from 6500 to 7500RPM) shift from first gear (acceleration) into second gear (for keeping up with other cars on the track), and select the third (overdrive) gear for best gas mileage, only when it makes sense (because the track allows large flat stretches of riding).
All these things also depend on what the track will be.
If there will be a lot of hills, you will want to find a gear that is able to climb the hills without losing too much speed.
If the parcours (track) has a lot of corners, you will want to have a gear that can accelerate fast enough, but once at a relatively fast speed, have a final gear ratio (or sprocket size) that will maintain that speed, at low revs.
If you are using a 4 stroke engine, you will want to rev at 2000-2500 RPM for best gas mileage. However, if the vehicle is too heavy, wind too much, or too many uphill roads, for the engine's size, you will need to focus in the higher revs, closer to where the powerband is, to keep up.
Still, to get best MPG, when riding is revving the engine as low as possible; don't aim the final gear for riding around in the powerband, unless top speed is important.
You get most power from the powerband (usually 6.5-7.5k rpm), but best MPG from lower revs (2.5k RPM).
More optimizations:
If it is any way possible, you can equip one of the wheels with an electro motor/genertor. If your track is one of a lot of corners and start/brakes, the electro motor could help out on accelerations, while store energy in the battery on braking (functioning as generator).
If the track is mainly long flat roads, with little cornering, hills, and brakes, then there's no reason to equip it with an electro motor.
If the race will be very small (say, less than 1 hour ride), there's also no reason to spend that much money into the vehicle.
Make the cockpit as simple as possible. Preferably the whole frame out of aluminum, and the plating out of plastic.
The rider's seat out of simple, light weight foam; don't go all heavy on a racing seat. KISS (Keep it simply simple), and keep it light weight!
The dashboard as simple as possible. It's better to mount the meters directly to the frame, than trying to make it nice, and install a dash board.
In the event of a fire, always equip the car with a fire extinguisher in reach of the driver's hands. Also a tool to cut the belt in the event of a belt jam.
Wear a lightweight helmet and preferably strong, non nylon gloves (in the event of a fire).
The brakes:
If you're mainly in a straight, even road, not a lot of braking or corners, equip the vehicle with only 1 max 2 disc brakes. Don't use drum brakes, they weigh too much.
If possible, use cable disc brakes. They are worse than hydraulic ones, but are a lot lighter, and easier to mount.
Use thin tires. Scooter wheels should be good enough (as they touch the ground only at a very thin strip).
The wider the tire, the more the rolling resistance will be.
Choose small tires, depending on the speed, and how it's mechanically possible, try not to exceed 13 inch rims.
12in rims would be your sweet spot.
Spoilers,
Keep spoilers down to a minimum. Spoilers are only good to add wind drag to give better cornering; but on long stretches, spoilers are bad (they add weight to the vehicle, and wind drag)!
Unless you have adjustable spoilers, that will flap up when cornering (like pro-F1 cars), but that's engineering on a whole new level (not a school project)!
That's about all the basic advise I can give you.
