Had a thought this morning as I was getting my morning cup of water. When playing a bard starting from level 1, when you level to 2 you get Jack of All Trades. Suddenly your character knows a little bit, about everything. Now if your game pauses between 1st and second level, and the characters train up to that level 2, then it would make a bit more sense.
But makes even more sense, to me anyways, would be if the Bard started with this ability at level 1. Like a fighter starting with Heavy Armor proficiency, the Bard's claim to fame is that he is a Jack of all Trades...but somehow he doesn't get to be that way till he goes and fights some goblins in a nearby cave?
Anyways, just thought I'd open it up to discussion, do you find this weird at all?
Leveling up in general is an out of character concept that is weird in general, but we've come to to accept it and explain it away in various ways, but for real it's not at all 'realistic' or anything. Maybe they've been studying the world as they traveled all a long, watching people, reading books, trying their hand at various crafts, and it just developed slowly but only became mechanically significant after some time. At level 2 it's only a +1 to all your non-proficient Skills anyway, hardly noticeable. Can be explained as simply becoming a little more experienced at being a dabbler.
"After that harrowing battle, the heat of the wizard's fireball still endangering your lute, your vision blurs for just a moment and something suddenly clicks! You feel a spark in your fingertips as new inspiration comes to you." --Me as a DM every time a player levels up and learns a new spell
A really good way to explain leveling up is having your character's going to a training area like an adventurers gym. This lets the DM have a little fun making like an Adventurers hang out. I once did this and also incorporated a shop where you can buy items and rations. This makes it more fun for the players and more believable as they level up. I usually put it in some obscure location like in the center of a volcano, you can even spend a campaign trying to get to it and then use it to level up. There's also sorts of things you can do with the idea of having an Adventurers hang out.