I have built more Derby cars than you can imagine, built them as kid, and then again for my son while he was scouts.
The big issue is the axle and keeping the nail in the slots, so I usually make my own axles and drill a hole to put the nail into. I also sand the nails using drill to remove the burrs and make them smooth. I would press in the nails into the holes and make sure to put lots of graphite lube, there is also white lithium lube which is powder, since you're not usually allowed to use oil based lubes. Before putting the nail into the axle I usually lube the nail and put the nail in drill and put the wheel on and work the lube into the plastic of the wheel, be careful not to over heat the wheel plastic.
Besides making sure the wheels are all on straight, make sure you have lube between the wheel and wood axle since this is other area o frictions.
Last you need to make sure the car weighs 5 oz. I use to go to the post office and get them to weigh it for me. I also set up the car so I can add or remove weigh at race day since their scales could weight more or less.
Here is where I bought all my derby car supplies
http://www.maximum-velocity.com/
We won most of our races.