Stairs are a bit intimidating. There are two basic types of construction. One is done with stringers and the stair components are attached on top to these, the other is know as "housed" stairs. Housed stairs don't have stringers. The risers and treads are let into the skirts and mated with each other with a T&G joint where treads and risers connect. This is the type that uses tapered mortices with wedges. This is your confusion, you are talking about two different construction techniques. A housed stair is completely assembled and then put in place where a stringer type stair is built in place, piece by piece. You don't have open treads with a housed stair.
When the rough stringers are installed, this is during framing, before sheetrock is installed. Typically, a 2x4 is nailed on the wall side stringer along the length of it's bottom edge and then the stringer is attached to the wall. This holds the wall stringer away from the wall studs 1 1/2", allowing sheetrock to slip behind it and later for the skirt board to slip in there, too. This gives you a continuous skirt board that doesn't have to be cut to fit against tread and riser cuts in the stringer. You have to know to do this early in rough framing since the underneath of the stair stringers usually get sheetrocked and you would have to remove it to move the stringer out. The skirt installs on top of the sheetrock.
If you didn't furr the wall stringer out and it's been sheetrocked under, you will have to cut a sawtooth skirt board. This is a lot more work and fitting the treads and risers to the sawtooth skirt has to be done gingerly or each will push the sawtooth out of line. Each of the teeth can flex and it can make the fitting of the risers and treads vary with every step.
To cut a skirt, you use a short piece of skirt board 1"x12" or 1"x10" to use as a template. You snap a chalk line on the wall to mark the top edge of the skirt. You want the skirt board to reach to the corner intersection of the riser and tread cut in the rough stringer and be the same top and bottom. The short template piece is held on the chalk line and you cut the angles to fit both the top and bottom. Once you figure those angles, you hold the template in place and put a tick mark on it and on the wall. Then do the same at the bottom. Measure the length between the tick marks. Hold the template on the full length skirt board, mark the top and the tick mark, then measure the length from tick to tick and position the template on the next bottom tick mark and transfer the angle cut. Easier to show than write about.
If you didn't furr out the wall stringer, you position the skirt board so it rests on the tips of the stringer teeth, which should be parallel to the chalk line you placed. Use a short level to plumb up at each riser, then cut a block equal in width to the amount of the riser. Place this block on the tread cut and scribe along the top to the skirt. This is done because the tread cuts may not be level and this marks the skirt cut parallel to the tread cut. If you get it right, the sawtooth skirt should drop into the cuts in the stringer and the top lands on the chalk line. You have a little fudge room since those cuts will be covered by the risers and treads as they are installed.
When you do the outside skirt on the open side, it can be held in place temporarily. Don't just scribe along the cuts in the outside stringer since these may not be perfectly in line with the other two stringers. Instead, use a straight edge held against all three stringers and use that to mark your cut lines. This way, the miters on the risers should fit the open side skirt miters.
When building piece by piece on stringers, you start at the bottom. Risers are typically in the 7 1/2" - 7 3/4" range but some can go as much as 8", particularly on utility stairs where room may be limited. This means you can't use standard 8" boards which are only 7 1/4" wide, you have to rip down 1x10s". Hold the risers level and even with the top of the tread cuts. This will allow the risers to provide continuous support for the front edge of the treads. Place the first two risers, then the first tread. You can reach behind and under the second riser to shim the tread if needed and when this is done, you run a couple screws through the edge of the second riser into the first tread, continue on up this way.
If you are going to use a scotia molding under the tread nose, you want 1" overhang on the tread nosing. If no scotia is used and the treads and risers will be wrapped with carpet or a runner, you only need 3/4" nosing but 1" is still OK. Risers get kicked and marked and treads show wear so it's recommended to use a runner or carpet. It also quiets down the stairs. Risers and treads can be nailed to the stringers, do not use adhesive since it won't allow the risers and treads to move with humidity changes and may cause a tread to crack open. I like to use trim head screws on the treads for greater holding. The holes are easily plugged with filler. Pay attention to where you put nails or screws on the open end so they aren't in the way if you are drilling holes for ballusters.
There is a lot more to it but this should get you started on the skirts.