For every mission you complete, you get Starlight Tokens, which can be used to craft the New Horizon Ward Skin or a Hextech Mystery Champion Permanent, which yes, is just a free mystery champion. It takes four tokens to make either of these.
To unlock the other missions, you must complete the first one. To complete A Star’s Beginning, you just need to play one game of the Invasion mode, win or lose. You get a single Starlight Token and a special summoner icon!
Each mission you complete after that simply rewards a Starlight Token.
The other missions are as follows:
A Guardian’s Origins
Win a game of Invasion: Onslaught with both the original and new Star Guardian teams as a premade group of five.
You have to win as the original team (Lux, Lulu, Poppy, Jinx, Janna) once and win as the new team (Ahri, Syndra, Miss Fortune, Soraka, Ezreal) once. You have to do this in a pre-made group. You also have to beat the harder mode, Onslaught, to complete this mission.
A Star’s Refinement
Win a game of Invasion Onslaught as:
Marksman: Jinx Ezreal or Miss Fortune
Mage: Ahri, Lux or Syndra
Utility: Poppy, Janna, Lulu or Soraka
You just have to win a hard mode game with one champion per position. (So you have to win three games).
HARD - A Guardian’s Trial
Win a game of Invasion: Onslaught as a premade group of five WITHOUT:
Marksman: NO Jinx, Ezreal or Miss Fortune
Mage: NO Ahri, Lux or Syndra
Utility: NO Poppy, Janna, Lulu or Soraka
Well. You’ll probably have to play three more games on the hard mode making sure you don’t pick any of the champions from each category. This part is a little bit hard.
HARD - A Star’s Ascent
Win a game of Invasion: Onslaught with a rank of “A” or higher.
This quest becomes unlocked after you finish one of the three other quests and gives TWO tokens.
HARD - A Guardian’s Pinnacle
Win Invasion: Onsalught as a pre-made team of five with an S or higher.
This quest is hard to accomplish, but you will receive TWO tokens (again) and a special Star Guardian icon.
As you can see, there is a bit of overlap for each of these missions. It shouldn’t be too bad to complete, as you have two weeks to finish them! If Invasion mode is giving you a run for your money, you can check our guide out here.
9/12/2017 Update: This article has been updated with a new information about “A Guardian’s Pinnacle.”