If you have a flat proof system in your tyres, like my ussual bike, I only carry a patch kit and a multi tool

If you can, carry a extra tube, multi tool, air pump, and a patch kit

A patch kit is:
What you need to dismount your wheels
Tires levers, 3 of them. Two for the work, one in case one breaks
Peel and stick patches
sand paper, make sure you upgrade from that scratching tool they mostly give you. Sand paper works better imo

A air pump is best a compact co2 pump. Online they are cheap. Make sure they fit your valve system and get extra cans. Carry three cans on a ride

You can also buy a body pump or clip on pump, yet they are a little cumbersome and won't deliver higher compression.

If you go C02, MAKE SURE YOU PICK UP YOUR EMPTY CANISTERS AND TUBES!!!!