I think you nailed it with the recommendation of a corded power tools. Over the years rechargeable batteries have improved but all of them will self-discharge over time and it is not uncommon for cells to flip simply because one cell discharged at a slightly faster rate. The end result is a battery pack that is useless when you try to use it after being in storage for a long time.

I have a half-dozen cordless tools that lack functional batteries around here. The batteries can be repacked with cells, I've done that a few times, but it isn't a DIY job for beginners and the new cells are not cheap. New battery packs are often more expensive buy alone than buying a complete kit with a new tool, two batteries, and a case.

On the other hand I have corded drills that date back to the 70s that are still kicking. I have a circular saw that was new in the early 60s. It has never been serviced and still cuts well. With a corded tool you have consistent power and performance without waiting. You have to have electrical power and a good extension cord is vital but once you have them you have something you can count on.

A simple 500w to 1000w, less than $100, inverter that will hook to a car battery can keep you working when power is out or otherwise unavailable. I have a 700w inverter than runs a 3/8" drill/screw gun, jigsaw, reciprocating saw, and 1/2" hammer-drill on small jobs. It lags with a 7-1/4" circular saw on long cuts and hardwood but for regular 2by4s and 1/2" ply it works well enough.

Makita makes a basic 3/8" variable speed drill that does most of what a homeowner might need for about $40. I have bought a few of them as backup units and second drills during commercial projects. They don't have the wrist breaking torque of a heavy-duty unit but they are more than plenty for 90% of DIY use. Add a 25', 14ga extension cord for about $20 and you have a tool that will still work ten or twenty years down the road.

For interior work a 25' extension cord is usually plenty. When the budget allows get a second. Buy commercial grade 14 gauge cords. If money is really tight, it happens, you might get by with short a 16ga cord if the tool isn't too power hungry. Light-up ends are worth the extra cost as they tell you when you have power.

Before buying tools I strongly recommend that you have a garage to store them in. A tool roll, toolbox, or soft-sided tool bag gives you a place to keep your tools. Keep them together and both clean and well oiled and they will last for many years. Without the central storage they wander off, get forgotten, and neglected.

A simple $5 voltage tester, a set of neon lamps with two short leads is worth the price. It will tell you when a circuit is live and what voltage/s are present. Gardner-Bender makes a nice little unit.

A box-cutter is a good tool. Disposable blades are cheap and expendable. I prefer the non-retractable sort, the Stanley 188 is a classic, but you have to keep it in a sheath or use the included guard.