Maintenance on a bike is much less than maintenance on a car. It is a much, much simpler machine with far fewer parts. As such there isn't such a thing as a maintenance schedule or service book.
That said, there are a few things you should do;
Tyres
Visual inspection and pressure check once per week.
Chain
The chain needs to be cleaned and lubricated. The frequency will depend on how much and how long you ride, as well as the conditions. I ride on average 100-150 miles a week and do mine roughly once per fortnight. In the winter when it rains a lot and there is salt spread on the road it tends to be weekly.
You should also check your chain for wear and stretching occasionally. You can buy a tool to measure the stretch. A decent chain will last anything from 5000-10000 miles or less if it isn't looked after.
You might also want to replace your cassette when you replace your chain, but this can be expensive. Again, if looked after a cassette should last 10000 miles easily.
Brakes
Check and adjust every so often, maybe once per month. Brake cables stretch and the blocks wear out and need replacing.
Gears
Once set up correctly gear mechanisms generally don't require much maintenance. On a new bike you may need to adjust after a month or so due to cable stretch.
Other components such as head sets, bottom brackets, wheel hubs etc are generally very long lasting and shouldn't require regular maintenance. In fact these mostly now come as unserviceable sealed units.
