I used to say that I just carried enough stuff to fix one flat tire and a good attitude.
Actually, that was never true. I also carry my bill fold, car keys, water and, for the last two years, a cell phone. On a new bike I pack along a folding allen key set until I get the adjustments dialed in to my satisfaction. After that I don't worry about it but the multi-tool generally stays in the bag until I need it for something else.
Tire inflation devices is a topic unto itself. There's basically two schools of thought. One says that, since you've already had a flat tire, you're likely to be aggrivated so you'll want a pump that's fast and easy to use. The other says that, since flat tires don't happen very often, you can live with taking longer to reinflate your tire in exchange for more convenient pump storage. Know yourself and choose wisely.
CO2 is, by far, the fastest and easiest to use but that comes at a risk. If you're not super careful in installing and seating your new inner tube, you'll blow it out and ruin your new inner tube in addition to losing your CO2. Mini pumps will do the job - eventually - but a guy my age might not have enough time left to inflate a 26 X 1.5" tire to 90psi. Full size frame pumps are about halfway between the two. A current hot seller is the Topeak Morph series which is actually a road side sized floor pump. That's what I'm using today.