Air
Check pressure with a gauge, not your thumb. The recommended range is printed on the tire sidewall; MTB runs lower for grip, road runs higher for speed.
Brakes
Squeeze both levers: they should feel firm and never touch the bar. Metal-on-metal noise means the pads are done.
Chain
It should look clean and run quietly. If it squeaks or is black with grime, clean and lube it before you head out.
Axles and bolts
Make sure wheel axles are closed tight and there is no play in the bars or seatpost.
Minimum kit
A spare tube or tubeless plug kit, a mini pump and a multitool. They take little space and get you out of most trouble.