Tires, if the tread gets thinner than 1/16" (and a lot are sold that way) one gets a lot more flats. I buy the tires with 1/8 " or thicker blocks and replace when lower than 1/16". tires older than 10 years tend to split the carcass, is the last flat I had, the only one this year. I ride >2000 miles a year and get about that out of the thick tires on road.
I leave my bike out in the rain a lot, so I oil chain, shifters & cables, brake levers & cables, deraileur pivots takeup and moving parts, front pivot, about every 3 weeks. Keeps it from rusting. Any steel brackets or wheels, need that also. cheap non-sealed wheel bearings need relubing anually. I use petroleum jelly on the wheel bearings.
Most critical fastener to keep tight is the front stem pivot, followed by the handlebars pivot tightener. If those are loose you have no control or stability. I've had a new bike from the LBS dump me on the ground after the stem was knocked loose by a chughole, so check that immediately after purchase. I tighten as hard as I can go on a 6" long wrench, maybe 50 ft lb torque? My new yubabike in January was fine on this point. New discount store bikes more likely to have a problem.
Most common fasteners to come off are the fenders. Axle bolts are prone to come loose, if you haven't double nutted them (counter torque) or used locktite those need checking monthly. Brake mounts and derailleur mounts should be checked then.
Spokes for a female or a careful male like me, need checking for tightness annually. tap them with a screwdriver and make sure all ring the same pitch, unless the tight ones (higher pitch) are compensating for a warped wheel. Thud is the wrong sound. If your wheel is too warped, or the tube is being poked by the loose and too long spokes, spoke adjustment is required immediately. I use a crescent wrench. spokes sticking up towards the tube more than 1/32" need filing or grinding down.
If the chain is popping over the rear sprocket, mid to higher gears only, either the takeup is too loose or the chain links are too long.