Are you buying a bike locally? If so, ask then to fill you in if they have any English speakers.
It is going to depend on the kind of bike you have and specifically what parts it has. Hub Gears vs. deraileurs, and the specific types of bearings it has. Here would be my set of simple repairs to learn:
- How to repair a flat, which involves installing a tube and a tire. Big issue is that you start with a package of parts that is easy to change. There are some nighmare combos that require massive force to dislodge, so if buying make it clear you want parts that work well together. Then get instant patches by Park, a quickstick, and a Road Morp type pump
- Learn how to adjust your deraileurs. Most of my gear is a little old school, so the shifters are pretty simple. You probably won't have to fix brifters (integrated shifters and road brakes), but if you do they are probably over the top. Deraileurs are mostly a mater of adjusting the cable length, and the limit screws. Screw some up, and learn to reset them.
- Learn how to break and refasten a chain. I carry a chain tool but there are links that come appart.
- Learn how to adjust your brakes. Another thing that is highly dependant on the type of brakes you choose. I would guess mechanical discs are the easiest. They work with the various lever types, and they have all kinds of adjustments for centering them. Do they squeak? Never had to worry with mine
I prefer Cantis, but they can be tricky with road levers. Vs can be tricky if the wheel gets out of whack at all, and need to be matched to special levers on road bikes, avoid travel agents (a device that makes normal road levers work with V brakes, because they are an additional complication)
- Bearings in the wheels and bottom bracket can last for ever or get the grease ruined early on. If you had the cash, Phil components are easily rebuilt, but cost a whack. Learning how to do regular bearings is easy enough but requires a lot of tools, a few of which may be heavy. On relatively dry and normal terrain you may not need to touch these parts.
- Truing wheels may not be required if the wheels are of superior build, truing wheel is a whole art in itself, there are books on wheels alone. I have several. You need a spoke wrench that fits your nipples. Small erors may only require adjusting the spokes nearest an outside bend, on the inside side. So if the wheel is off center left, make very minor adjustments to the spokes on the right. You can use the frame or caliper brakes to indicate wheel trueness.
- Cranks will often come loose on tapered bottom brackets on new bikes at around 500 miles. The cranks start clicking. If that happens find a store to pull the cranks, lube them and reseat them, or hit them with a big wrench. Pedals also need tightening with a big wrench.
Bikes are not that complicated though they can be subtle. Just look at every part of your bike and think about how it works. You can find online tutorials for just about anything.