Nope, everybody starts trained at the stuff their class does, The warrior his weapons/armor/etc. The Wizard his magic, etc. The Thief his thievery etc.
But your Warrior CAN learn to pick locks and get better at it given enough tries. And your Wizard can become proficient with a crossbow given enoug tries.
What I want to do is to allow your character to grow in-game and not off screen.
So you want to learn to play the lire? Fine, role play it for me and then if you do it well you get to roll and I might even give you a bonus for your roleplaying.
It doesn't matter you're a warrior, it matters that you invest the time in-game to learn it.
Ok I see more clearly what you are getting at here.
So maybe something like this:
Everything would be an ability check. So every class can theoretically do/try every "skill".
(Magic users would be considered 'gifted' so not everyone can just learn magic - there would be some limits - and there would probably be a few more I can't think of now.)
But PC's get a bonus for their "class stuff": The warrior his weapons/armor/etc. The Wizard his magic, etc. The Thief his thievery etc.
So essentially each class has a list of a "special set of skills" that they are proficient in.
I would simply say that they can announce that they are training for the skill and will have to learn it in down time.
So they have to either spend a certain amount of XP over time - or you make a downtime training mechanic that for every x days spent training they get to roll a d6 and mark off the amount of effort spent to learn a new proficiency. When they have invested in enough training time to mark off enough effort, they then get to add that skill to their class proficiencies.
Personally I would keep it as simple as possible.
+2 Student
+4 Expert (All classes have some proficiencies that start at this level)
+6 Mastery
In this way you can make all proficiency increases have to take place under the downtime mechanic, or maybe have a mini system where if they roll a natural 20 or beat a DC by 10+ they get to mark
x amount of effort towards advancing a proficiency.
The higher the proficiency level, the more effort they need to mark off to advance it.
I would really want to avoid any system that does a bunch of small +1 Increases, as I feel it would defeat the purpose of getting rid of the complication of a skill system.