The ultimate choice for Zombie Survival transportation!
Second, bikes are essentially silent, while cars are loud. Go stand next to a freeway and you will instantly see how loud a car is (and hundreds of them especially). Since noise attracts the undead, the quieter you can be the better! A properly maintained bike will cruise along without creaks or squeaks, carrying you in silent safety from the undead.
As alluded to above, bikes are generally quick and easy to maintain. If you know how to fix a flat, lube your chain, change/repair the chain, and have a few spare parts, a solid steel-framed bike could last you basically forever. Best of all, all the tools for this would weigh no more than a few pounds and many could be easily found as well. Didn't stock up on your chain lube? Just find a broken down car and drain the oil and use that!
For a basic kit, I'd suggest some tubes, tires (you can get nice folding tires now that hardly take up space), chain lube, chains, a basic multi-tool, a nice floor pump that fits both presta and shrader valves, and a set of hex wrenches (or use the Park 3-Way Wrench, which could double as a close-combat weapon!). If you have a group of cyclists, you can easily split up the kit and have an even better supplied tool kit with which you can fix just about anything. Plus, you can use many of the same parts on different bikes, eliminating the need to carry too many extras.
Next - you won't get stuck on a bike like in a car. Again, watch any zombie movie or TV show (The Walking Dead is a perfect example) and you'll see the group of survivors get stuck on a freeway crowded with abandoned cars. This would never happen on a bike, where you could easily weave around the stalled cars. In addition, you can easily take the right bike off-road. Any decent mountain bike will give you a huge advantage over a car. No matter how well a car is built for going off-road, a bike is better prepared and can go more places. And on a bike, if the going gets too tough, you can get off and push or carry it.
Lastly, while most people suggest that a bike is limited by what it can carry, that's definitely not true! You can easily outfit your bike with panniers or a trailer to carry a ton of stuff or choose one specifically designed for the purpose like an Xtracycle, Surly Big Dummy, or a cargo bike! A cargo bike isn't quite as adept at going off-road and can't be carried as easily, but it's still much more versatile than a car, plus has all the other benefits discussed above!
The folks in Portland have outlined the great utility of the bike for disaster relief and many of the same principles would apply to a zombie outbreak: http://bikeportland.org/2013/07/13/disaster-relief-trials-bring-30-miles-of-urban-apocalyptica-back-to-portland-90373 So if you want to be the best prepared for the undead, forget your car and outfit your bike! As a special benefit, same applies for your daily commute or running errands - your bike is often the best vehicle for the job!