The term you are looking for is a wood "plug." You can either cut a piece of dowel, or if you want control of the grain direction, use a plug cutter to cut your own. A flush-cutting handsaw can be helpful, as well.
Regarding screws, you won't find a definitive guide, because the strength of a joint depends on what the joined object will be used for. You can make a 1/4" thick triangle without a center into carpenter's ruler for measuring or to be used as a shelf support.
I believe people learn what size screws to use by starting out by learning from others (these days, like watching YouTube videos). With that foundation, over time you will discover when screws have proved insufficient or when it didn't seem to matter (which meant they were sufficient).
It is not necessarily a bad exercise to screw (and glue) some pieces of wood together and try to bend them apart. This will help give you a sense of when strength depends on the choice of screw (or the choice to use screws at all).
You may discover that under stress the fastening method remains strong while the wood is delaminating/tearing apart. Keep in mind the direction of the forces at the joint, as well. Screws are incredibly strong against a force trying to pull wood apart along the axis of a screw as well as a force 90° off the axis (trying to "cut" the screw in half), but stress is rarely in a single linear direction. Predrilling a hole before putting the screw in can be a part of the learning process as well. You can find guides for how big the hole should be.
I believe that ultimately a good decision can be made after one becomes able to consider the kinds of stresses that will be put on the joint (combined with how much the structural integrity of the wood will be lost by putting a hole in it).
Fortunately we can partially rely on the fact that the common gauges of screws available fall into a narrow range depending on how long the screw needs to be. You can trust that an intermediate choice is likely to be fine under normal circumstances.
If the stresses may be unusually high, consider that the design of the joint may be the limiting factor, not the screws. Keep in mind that one can make strong joints without using any fasteners or glue at all (see Japanese joinery for an abundance of information). The best screw is not going to work if the joint is structurally too weak for the stresses it will be subject to. Ask yourself, what is the purpose of the screw?—when properly applied glue will be stronger than the wood itself.