Advancing Skills in Game
Just some observation, personal preference, things to think about, and discuss.
Learning by doing
I can see some realism in a system based on learning through use. Is it impossible to learn something (perhaps a new level 0 skill) through normal everyday non exceptional practice? Is it not possible to learn from one's mistakes? Not sure why a system should be based on just box cars or exceptional success.
This type of learning by doing seams to me most likely, or easiest, at the lowest levels for many skills. Still plausible even at the highest levels as the innovators who push the bounds of knowledge and abilities do so via their own hard work and research and not by just duplicating what others know.
Pick up a level 0 skill through practice and experimentation. Move up to level 1 through use and research and so on.
The higher levels would probably be more difficult and time consuming than learning through other methods.
Self taught
Instructional books, on line learning, simulation machines and so on. Learning things without an instructor and at your own pace is certainly realistic. For many skills, this includes some learning by doing too.
Obviously there is a limit of learning based on the level of the resources available. Some skills will need the proper equipment and resources for simulating or performing "real world" tasks as things like Medic: stitching up a cut or Survival: starting a fire rubbing sticks together, are best learned through actual practice and are not just learned in a book.
Learn from a mentor
It is certainly realistic for someone with skill to teach what they know to another. This method likely includes some learning by doing with supervision as well as some practice on ones own. This method can include some directed self teaching "Here, read this" with help when needed "I'm having trouble with the section on reducing transmutation decay".
Learning with assistance should certainly go faster and be less difficult than self taught or learning by doing alone.
Learn from an instructor
This type of learning is dedicated time spent learning from someone practiced in instructing the subject matter. It likely incorporates teachers aids "Learn from a mentor", research assignments "Self taught", and practice labs "Learning by doing".
To me, it may be a lot of dedicated time and hard work, but ultimately the quickest, safest (want to learn jump engineer 2 through trial and error?) and most likely (if basing success in gaining a skill on a task roll, this would have a lower difficulty than, for example, trying to teach oneself Medic 2) way to gain skill, especially at higher levels.
Advanced learning methods
With advances in tech levels, advances in learning may also be possible. Perhaps these methods still use whatever game mechanics one utilizes for the less advanced learning methods but there is a modifier to the time required and/or the difficulty.
Take the good with the bad
Any system that allows increasing skills should include some balance. What are the options for losing skills due to lack of use? Old age? Improper training? Learning to cut corners? Bad habits?
Pitfalls
Base learning by doing on task results and players may seek out opportunities to roll instead of role play.
Realistically, some skills may take longer to learn than others like Medic 1 vs Vacc Suit 1. Some skills, like astrogation, may be easier to learn from books or on line while others, like art(sculpting), may require more hands on work.
Get too complicated and a system can turn into a lot of record keeping that distracts from role playing.