this is two questions really;
The answer to the first is yes - all engines - whether bikes, cars, tractors etc have a "run in period" where you should limit the RPM and speed of acceleration until all the parts are fully bedded in. Each manufacturer will have their own run in period.
The answer to the second is you should learn the following:
- Chain/Belt drive tension: ensure you are able to correctly check and adjust the tension of the chain/belt drive - a loose chain/belt can cause excessive wear and/or break
- Chain lubrication: ensure you know how to clean and lubricate a chain (if fitted)
- Oil level and maintenance - some bikes require you to check when the oil is warm, others cold, some on the side stand some vertical. Check your manual for details
- Brakes: check there is no debris in the brakes and that they operate smoothly.
- Lights: regularly test all your lights, preferrably before each ride
- Tyres: inspect them for cuts, debris in the tread and wear before each ride
- Keep your bike clean: its easier to notice things that are not right if its clean; secondly if you regularly clean it you will have to look at all the bits up close regularly and be able to spot any leaks/breaks/wear more easily.