The language around mortise and tenon joints can be confusing, so it may be helpful to familiarize yourself with the common names for each part.
In this example, we are making a blind mortise (partial depth, as opposed to all the way through) and a shouldered tenon with four shoulders (as opposed to barefaced). Mortise and tenon joints can be designed in any number of different ways to suit different use cases and resist different forces.
Start with flat, square, true stock. Plan your mortise carefully. In most cases, the tenon should be about one third of the overall thickness of the stock, to balance the strength of the tenon and the walls of the mortise. If the mortise is located at the end of the workpiece, the shoulder must be 1/2" or more to avoid weak short grain. Hold the work in a vise while you lay out the mortise using a combination square and a knife. Make the knife line with multiple passes, starting with very light pressure. Pencil mark the depth of the mortise and its center line to help set up for drilling.
At the drill press, choose the largest drill bit that will fit within your mortise dimensions. Set the depth stop so that you consistently drill to the correct depth. The cone-shaped tip of the drill bit should extend below the depth line--or better yet, use a brad point bit.
Drill each end of the mortise first, then as many holes as possible in between. Drilled holes cannot overlap, as the bit will deflect and wander off course.
Secure your work to the benchtop with clamps so it is well supported. Staying ~1/16" away from the layout lines, use a chisel and mallet to chop out the waste left between the drilled holes. Keep the walls vertical.
With the bulk of the waste out of the way, use the widest chisel that will fit to carefully pare the "cheeks" of the mortise. Take light cuts, working outwards to the lines. You can feel when the chisel registers in the knifed lines, so you can be sure of accuracy. Use a square to help you see that the flat back of the chisel is staying vertical. Work back and forth between the long grain and end grain faces, aiming to get a clean corner. Use a narrow chisel bevel-up to level the bottom surface. Clear out any crumbs and sawdust, as they could prevent the tenon from seating all the way.
Use a combination square to assess the mortise. For the long-grain "cheeks," pinch the ruler against the inner wall surface and the anvil against the outer surface of the board. If they are parallel, the cheek is cut correctly. If the cheek leans out, keep paring. If you cut too far, sometimes you can glue in a thin scrap of wood and then pare it back.
For the end grain cheeks, register the square on the face of the work and slide the blade up to the cheek. If there is a gap at the top, keep paring. If the blade makes contact all the way along, the cheek is square.
Be picky! The better the mortise, the easier it will be to fit the tenon.
Choose how the parts will be oriented to each other and make a "marriage mark" across both so that you can repeat this position. Layout the tenon dimensions from the mortise, then carefully carry those lines across the edges and faces with a combination square and a SHARP pencil. As always, the stock must be square, flat, and true.
Layout the length of the tenon by cutting the shoulder line with a knife and combination square. Take multiple light passes for each line. Cut one face, then carry the line around the corner to the adjacent face. Register the knife in the previous cut on the corner, then carefully slide the square's blade up to contact it. Pinch the square in place, then cut the line. Repeat for all four faces (assuming the tenon is shouldered on all sides).
You can cut the shoulders with a handsaw, the bandsaw, or the tablesaw crosscut sled. If you use the tablesaw, carefully set the depth of cut (it may not be the same for all faces!) and the alignment of the shoulder. Use a stop-block clamped to the sled's fence so that you can repeat the shoulder position for each face.
You can cut the cheeks with a handsaw or the bandsaw. If you use the bandsaw, check that the face of the fence is square from the table surface. Align the fence with care, and stop if you see that the cut is deviating from the line. Cut all cheeks.
Show the parts to each other. Ideally, the joint will go together with minimal slop and no gaps straight off the saw, or with minimal fitting. Aim for a good, snug fit that goes together with hand-pressure or light mallet taps. You should not have to drive it together with a sledge.
To pare the cheeks (long grain), work cross-grain with the flat back of the chisel registered against the cheek. For the shoulders (end grain), register the chisel in the knife line and pare with a slicing motion. Always secure the work in a vise!
Take care to determine the source of any issue with the fit--don't just start cutting at random. Use the combination square to check for parallelism and squareness.
To help the tenon start into the mortise, lightly chamfer the end. Use a chisel skewed towards the endgrain. This also creates a little room for excess glue inside the joint, and prevents any remaining crumbs or sawdust in the mortise from holding the joint open.