Under Magic
For one, you join the College of Winterhold. Well, your currently in the biggest (and only) magic institute in Skyrim, so what do you do? You run into dungeons, firebolts flying and (possibly) axe swinging. As a first school project.
Two, training is very individual. They should be retained as an option, but you should be able to attend some classes (possibly just some short dialog and a time skip) that advances your skill, at the cost of a notably smaller sum of money, and limited hours. Or put a division in the speed at which you advance. Individual training should advance your level more (don't agree with just an instant skill rank-up) and have the "training limit" turned into available hours.
If you want realism, have some calculations between the skills and your stats (also kinda like the return of stats, just not micro-managing skills all the time) (instead of just your level*gold) to create a general understanding value.
For the personal trainers, with your higher grasp on magic, you would require less time to advance, giving you the option to grow more rapidly in less time, at a higher cost. Add the option of apprenticeship, where the trainer will not charge you for training, but will have some alternate demands instead.
For the class courses, with a higher general understanding, they will give you a "scholarship" that gets progressively better with your aptitude, ending with (possibly) a class 'A' scholarship that completely negates the tuition. There would be a limit of how high you can go with just classes of course.
Under Combat
Enable the use of practice dummies to some degree.
The ability to study under a master with multiple students. You will get some individual focus, but not as much as you would get under private training. They will mostly study and comment on your stance, and occasionally make you spar against them. Eventually, you will have to get some private training where you would just spar, instead of making use of dummies.
Under Stealth
Hard to guess, but I would say all of this would have to be individual, given the less-than-legal nature of many of the skills. You would have someone that you would pay to pretty much follow you around as you practice your skill (within a certain distance of where they were hired), and occasionally stops you between activities to correct your methods. If you get arrested, you can pay your fines, or they may be nice enough to visit you in jail, offering to help you get out (based off of their individual skill). Ex: the lockpicking trainer would be able to slip you some more picks as you attempt to break out. The speech trainer would help you talk your way out or distract the guard for you. A barter trainer may be able to help you get a reduced fine.
They all may be wiling to pay your fine for you now, but could ask a favor or reimbursemant for it later, maybe allow a trade in for payment? ex, you find a few items, and "trade" them to the trainer, who will then assess the retain value of the wares, and give you any excess money back, keep the money as a tax, or tell you that you need more. (dependent on disposition, skill, and the cost of the fine that they paid for you.
Other?
Also allow for the development of combat styles. I've developed into an assassin-like character, so why do I swing my dagger/shortsword so haphazardly. My methods would be quick accurate strikes with the goal of hitting unguarded or vital locations.
Maybe I make frequent use of poison. Wouldn't I learn eventually where those poisons have the most crippling effects? Frost Oil on the legs could greatly impair movement, shock oil towards the head/chest could cause seizures (paralysis) or stop their heart. A slow release magicka poison in their hands may prevent the flow of energy, impairing the ability to cast spells.
I have a style devoted to counter-attacks and defense more so than first blows. More than likely, my biggest attacks would be lunges (with a sword) after blocking. I wouldn't drop my defense so much as to perform an over-head arc.
If I fight like a berserker, with only the intent of killing my enemy before they can act, then I probably would focus very little on defending, and prefer powerful, but somewhat quick blows. I might rip through the stamina noticeably faster, but I should, hopefully, kill you before that's an issue.
Maybe I'm a spellsword. I would probably keep a tighter style, meant for keeping my movements balanced and quick. Using my magic to create a breach in defenses (magic with a stagger %?) and using the openings for jabs, thrusts, pommel bashes, or an infusion of my magic and blade in the form of a powerful sweep.