I would say it depends on what kind of riding you are going to be doing. Shortish recreational rides? Loaded touring? Off road singletrack?

At a minimum I would carry a patch kit and/or spare tube (or 2), tire levers and an inflation method, whether a pump or CO2 canister. The most common issue you will run into is a flat tire.

I have an old version of the Alien tool and it works fairly well. The large majority of issues will be tire related or can be adjusted with an allen/hex wrench selection. I keep mine in an under seat bag so it is out of the way. Depending on how much gear you want to carry, there are racks/pannier options, trailers for tons of gear, or if you are off-road you may look into bikepacking gear.

I have never needed to break a chain on the road, but have done it several times at home (installing components, etc.). The chain repair links weigh nothing so they are nice to haves and make life easy if they are already on your chain. Most bike specific multitools will also have a chain breaking tool.

If you are touring I would take spare spokes or a fiber fix temporary spoke. Tire boots are ok but rarely needed and can be improvised. Also for touring spare brake and derailleur cables don't weigh much and can come in handy. Not really necessary for short rides.

Get a bottle cage and always take water. That's a no brainer. On my touring bike I have 3 cages and bottles. Camelbaks work too but bottles are more versatile.

Most importantly I would take a maintenance class and learn how to work on your bikes. I took one many moons ago and learned a great deal. I have built the last 6+ bikes from the frame up. That way you know how to fix it if something breaks. REI offers them for free for members. I took mine through an adult ed class. Check with the bike shop you get your bikes from, they may offer a class.

Enjoy your new rides and have a great time!