Sports typically require a long zoom lens.
Narratives do great with wider ranges and you have lots of time, so primes give you better quality usually for less money.
Kit lenses, while versatile and cheap, are usually slow and the image quality isn't great.
Finally, you're on a crop sensor, which is important to consider.
All that said, with what you want to do and your budget, you'll have to compromise somewhere...
Lens quality compromise:
Pick up a Canon 18-135mm kit lens for $300ish. It's VERY versatile: can get wide on your crop sensor and relatively tight at 135mm. However, it's slow. You'll want some stadium lights on at night for sure, but daylight will function great. This was my first lens (came in the kit) and I still hang on to it today for those weird situations where you need an "everything" lens. Alternatively, you can shop for third party zooms like Sigma or Tamron. I'm not a fan of either, autofocus is slow and image is worse than the canons, but man, they're cheap. (Don't get a 24-105 for a crop sensor everything lens. 24 is too tight. You'd be happier with a 17-40 for the same price).
Versatility Compromise:
You can pick up a really high quality Rokinon prime lens in your price range. MAYBE two if you shop around a lot. Their facebook page posts deals every now and then where some retailers sell them for over half off. They're fast (work great in low light) and beautiful. Perfect for narrative film. You're going to have a horrible time shooting sports though, as you'll have to swap lenses to change focal lengths.
Also, if you want a quality lens for sports but not so versatile for film, you can pick up a Canon 70-200 f/4 used in that price range. Another beautiful lens, but you'll have no wide end. Again, that's fine for sports and talking head stuff, but yeah... I don't recommend this as a first lens for anything else.
Budget Compromise:
If you can stretch your budget, you can pick up the longer lens (70-200) and wider prime (fixed but good for low light) or zoom (17-40, still versatile but needs more light) or all three.
Auto-focus compromise:
You mentioned autofocus, you typically don't use that in pro video. If you can get around not having it, buy vintage lenses. You can pick up an M42 mount 200mm for $40, a 135mm for $80ish, 50 for $50-100, 28 for $20-50 and even a zoom for a couple of hundred, plus your $20 adapter. Vintage lenses have a look all their own, but it's a pleasant look. They're less expensive and hold quality.