You get a small amount from the "First Win Of the Day" quest, but the majority of your BE is supposed to be coming from selling champion shards you get out of the capsules you get whenever you level up your account. Depending on the champ(s) you pull, you can get a few hundred to a couple of thousand BE off a single capsule. A few levels can easily get you almost any champion you want.
You can also get a sizable amount of BE from events and Hextech crates. Those tokens you earn from events like the Odyssey event can be exchanged for BE, and many of the event missions also reward BE. When you earn those Hextech keys and chests from playing and being a good sport, you can use the champion shards you get.
For any new player, I highly recommend treating the game as free-to-play only as a trial run; once you've decided that League is something you want to invest your time in, put $5-$10 bucks down to get a champion that suits your playstyle, and use that champion to have an enjoyable time grinding towards whatever else you want. That amount is still far cheaper than most games out there, and for the amount of content Riot makes and the hours you can put into League, it's well worth it.