First win of the day, which gives a chunk of xp and a bit of BE.
Missions/quests from events tends to give event tokens, which are usually redeemable for stuff like chromas/key frags/special borders more so than BE. Theres usually a convert to BE option for any leftover tokens, but it's like 10 BE per token (and we're talking <100 tokens), so it's not much.
Leveling up and disenchanting the resulting champ shards is the primary way to get blue essence, which is why xp boosts are the way to go if you want to pay for speedy BE. (Although if you wanted to pay, you could just buy the champs you want directly with RP.)
Since you seem to be a legitimate new player, just focus on trying a variety of champs (i.e. try each of the free week champs) and learning the game (via playing games), and don't worry too much about getting specific champs. If you played someone on free week and they really resonated with you sure you can save up for them, but it's easy early on to see a champ and think they seem really cool/powerful, only to eventually play them yourself and have a rough time and end up not enjoying yourself.