Andrew, first, decide on the appearance you want. Full overlay doors or partial overlay (where you see some of the face frame between the doors). Next comes the hinges. Most everyone these days use fully concealed hinges so mounting becomes a consideration and can dictate cabinet construction.....frameless or with a face frame. The face frame stile width will be decided by the hinge and overlay amount.
In different cabinet shop, we used mortise and tenon as well as half lap face frame construction. The half lap is much easier/quicker and the most strong.
The current trend of pocket screw construction is fine but in my book it is still a butt joint....not as strong as other joints.
For cabinet box/carcass, 3/4" plywood for frameless and 1/2" plywood for framed construction. I always include a cabinet back. Usually 1/8" Masonite or plywood. Dados and rabbets were my preference.
Standard vanity cabinet depth is 21" with 18" for a shallow vanity. The tops are usually 1" overhang 1" or more with full overlay.
Toe kick 2"x 3" years ago. Now, I would go taller....3 1/2"- 4" (kitchen cabinets are normally 3"x4". Cabinet height is what you want now. Instead of traditional 30" high, most are not up to 34" high. (Standard kitchen is 34 1/2").
As for doors and drawer fronts, if you don't have the tools, skills or desire, you can order these from shops that specialize in these products. Then you just make the cabinet.
Remember, accuracy in planning and measurements will make the job go much smoother.
If you have any questions, shoot me a pm and we can talk by phone.
Good luck!