I'm making a druid and drawing up her backstory when I realized some details might conflict with the mechanics of the game. For example, I initially had the idea of her discovering that she could turn into smaller animals (think mice, snakes) with the help of her village's druidic mentor growing up, but mechanically speaking, druids gain the ability to wildshape at level two, and as I plan to multiclass I might be taking two or three levels of cleric before finally taking my second druid level, meaning that my wildshape may come into play a little later into the story.
Which is fine, it just made me reconsider including it in her backstory since it seems redundant to have discovered it already without being able to use it within the rules of the game. Of course this all depends on what level your DM will be starting the party out as, giving you some room to be a bit more flexible with when they would have discovered it, but I was just thinking about it flavor-wise. Say I forgo her discovering the ability at a younger age and instead have her discover it a good ways into the game, how would you go about it? From what I've seen a lot of people tend to hit level two and immediately take to wildshaping as if it was second nature, but I'd want to roleplay it out in a way that's true to character as she's still young and discovering her druidic abilities.
Forgive me for my lack of imagination, I've only played a handful of sessions and haven't seen many examples of how one could go about actually discovering their wildshaping abilities. Would you describe it as an innate power that you can just tap into once you realize it's there, or maybe play into it like lycanthrope and have the first wildshape occur involuntarily, forcing you to learn how to control it? I'd love to hear some ideas on how you think I could go about it! 

What I would do is modify the backstory a bit.  Your mentor taught you the process to shape change, but you never fully mastered it.  Have you seen Star Wars and The Empire Strikes back?  In the first movie, Obi-wan starts to teach Luke about the force.  He's able to do small things, but he can't really count on his force abilities.  In the second movie, when he's captured by the wampa, he /NEEDS/ to get his lightsaber so he can escape.  He tries to use the force and it fails.  Then, he calms himself, focuses, and is able to pull the lightsaber to him with the force.

Something similar could happen for your character.  Maybe even roll it out.  You know /how/ to shapeshift, you just haven't passed the mental hurdle yet and actually done it.  Self assign yourself a DC, maybe it's even a harder check like a 15 or a 20.  When you want to wild shape, you make a wisdom check.  If you fail, you don't shape shift.  If you pass, BAM you're in.  Now that you've done it, lower the DC.  Next time you try, now the DC is 10 instead of 15.  Roll, and if you pass, great!  You're in!  If not, phooey, you need more work.  Keep trying until you successfully pass.  Then lower the DC to 5.  Once you pass that check, you've practiced enough in combat situations that you've mastered the skill.

There might be a few different ways you could go about doing it depending on your character's personality.

• One idea that you might have to ask the DM for permission would be that she did have a mentor growing who tried to teach her to wild shape, but she could never get a handle on it. She still attempts to try to wild shape from time to time (prior to getting to Level 2 Druid) but it still always fails. (This is the part that needs DM permission) You could have her try, during moments that it would be useful, to wild shape but she can only get a partial transformation. Tries to turn into a cat, gains a tail and fur. Attempts a squirrel and gets the ears and teeth but nothing else. So on and so forth until she final gets the trick (Level 2 Druid Whoop)
• Another idea would be sort of stealing from the Warlock class. During some long rest in a forest, "the Spirit of the Forest" comes to her in the dream and grants her the blessing of Wild Shape. This might be dependent on the campaign story but again would require some DM assistance. 
• Then there is the idea that you use to be able to Wild Shape when you were younger, but due to some trauma when you were in a mouse shape blocked your ability to use that ability. So it would require you to Roleplay out that difficulty and overcoming it. 

But there are a few ideas for you if any of them seem to your liking. And I look forward to anyone else's ideas.

Quote from crzyhawk >>

What I would do is modify the backstory a bit.  Your mentor taught you the process to shape change, but you never fully mastered it.  Have you seen Star Wars and The Empire Strikes back?  In the first movie, Obi-wan starts to teach Luke about the force.  He's able to do small things, but he can't really count on his force abilities.  In the second movie, when he's captured by the wampa, he /NEEDS/ to get his lightsaber so he can escape.  He tries to use the force and it fails.  Then, he calms himself, focuses, and is able to pull the lightsaber to him with the force.

Something similar could happen for your character.  Maybe even roll it out.  You know /how/ to shapeshift, you just haven't passed the mental hurdle yet and actually done it.  Self assign yourself a DC, maybe it's even a harder check like a 15 or a 20.  When you want to wild shape, you make a wisdom check.  If you fail, you don't shape shift.  If you pass, BAM you're in.  Now that you've done it, lower the DC.  Next time you try, now the DC is 10 instead of 15.  Roll, and if you pass, great!  You're in!  If not, phooey, you need more work.  Keep trying until you successfully pass.  Then lower the DC to 5.  Once you pass that check, you've practiced enough in combat situations that you've mastered the skill.

To lean into this further, maybe you have manifested the ability to WS but you have no control over it, so you can't do it at will. Would make for some outstanding and hilarious RPing if the DM triggered such on you outside of combat at random times.

This is a very common theme in movies and books for individuals with inate abilities. Should be able to find some good examples to model.

Building on what others have said about knowing the basics but not having mastered them, feel free to throw in some flavor.  Your character spends her free practicing shape changing.  Sometimes she can sprout a bunch of hair on her arms or make her incisor grow a bit.  Did she take the primal savagery cantrip?  Link that to your attempts to wildshape.

One idea that you might have to ask the DM for permission would be that she did have a mentor growing who tried to teach her to wild shape, but she could never get a handle on it. She still attempts to try to wild shape from time to time (prior to getting to Level 2 Druid) but it still always fails. (This is the part that needs DM permission) You could have her try, during moments that it would be useful, to wild shape but she can only get a partial transformation. Tries to turn into a cat, gains a tail and fur. Attempts a squirrel and gets the ears and teeth but nothing else. So on and so forth until she final gets the trick (Level 2 Druid Whoop)

Also she could get smaller and bigger but not the precise size.

Good ideas from others. Maybe while playing in the woods you get corners by a dangerous animal and in fear for your life you transform to escape. Afterward that’s when the mentor appears, your transformation drew their attention and though they try to teach you, you cannot shift. It’s only after much training (level 2) you get control and can use it. 

A lot of variations you can use without having to scrap your backstory 

I'm a big fan of primal savagery as your proto-wildshape in keeping with your backstory.