I just carry a patch kit like this one and some levers like this. They go in a saddle bag, and that's all I generally need. Add a multi-tool if you want to adjust things as well.

I usually buy this kind of stuff at my local bike shop, which is usually a little more expensive, but I think it's worth it.

I’d say it depends on the type of bike and tire.

Tubed or tubeless? For tubed, I’d recommend 1-2 spare tubes put in a saddle bag or one of those fake bottle kits that mounts in a bottle carrier, a small co2 pump, a few cartridges and a tire lever if yours are not easy to remove and install by hand. If you have a flat, replace the tube, then patch old tubes at home. For tubeless I recommend the dynaplug tool, optionally a small bottle of extra sealant and a valve tool. Small hand pump as co2 can interfere with tubeless sealant. I ride tubeless and just chuck these items in a jersey pocket.

I usually outfit my bikes with a bag under the seat which includes all tools for changing a flat and a spare tube if there’s room. Topeak makes a few that come with the tools included but if you have the tools there are plenty of small bags to choose from. Also, don’t forget a frame mounted hand pump (if you’re not using CO2).

If you're a beginner just pack a spare tube, a hand pump, and a tire lever. Maybe a patch kit if shit really hits the fan and you get two flats on a ride. But you don't want to rely on patch kits.

Don't mess around with CO2 unless you're in a race or don't have room in your kit for a pump. C02 is fast and small but isn't fault tolerant.

Make sure to practice tire changes at home so you know what to do if you get a puncture.

Most importantly make sure your tire is at the correct pressure (check weekly), most punctures are due to pinch flats from under inflated tires. I've put in 1000s of km and never changed my own tire, but I've rescued plenty of strangers who rode on under inflated tires.