#1 is keeping chain clean and lubed. Relube when it starts feeling dry or noisy and after rain rides (for me ~100mi). Relube and run chain through rag to remove dirt and excess. About every 400-500mi or so I use a chain machine cleaner and gear floss on cassette to to bring everything back to a like new state of clean. I'm kinda OCD with my gear though.

Give bike a good wipe down with a damp rag when it looks dirty; inspect frame for cracks once in awhile etc. Make sure wheels are spinning true, spokes feel tight as new. All bearings running smooth? (i.e. headset, Bottom Bracket, Hubs) Occasionally you may need to tighten a barrel adjuster, or turn a limit screw to tweak shift indexing or to compensate for brake pad wear. I ride Speedplay pedals, so that means periodic cleat lube and spindle greasing, as well.

Other than that as long as everything feels good and sounds normal you're good to go. I mention sound because the majority of mechanical problems will make some type of noise, i.e. bottom bracket creak, headset clunk, noisy shifting, screaming if brakes fail (just kidding) etc.