The big chainring is probably supposed to be like that, unless all the teeth are worn down. On modern chainrings, some teeth are shorter to aid chain shifting.
If your tires do not have cuts or are not cracking, they are probably OK. If you see a lot of cuts or the tires are very thin and cracking, replace them.
For the chain, measure 12" from one link rivet. It should be exactly 12" to the rivet at the 12" point. If it is more, even a fraction more, replace the chain. If the new chain jumps on the cassette, replace the cassette.
Check for play in the bottom bracket by grabbing a crank and trying to move it from side to side perpendicular to the frame.
Check for play in the wheel bearings by holding the rim and moving it gently from side to side.
Check for play in the headset by applying the front brake and pushing the bike to and fro while feeling for play around the bearing cup just above the fork or below the handlebars.
How does the bike feel when you ride it? Are there any unusual noises?
Lubricate the chain once per month; check tire pressure before each ride; check condition of tires before each ride; clean chain when it appears really dirty. This will depend upon your riding conditions.
There's more, but don't become obsessed. Enjoy riding!!