Since I did a similar threads for the Spirit Blossom and K/DA events I'm just gonna continue the tradition. Hopefully someone will find this useful.

List of regular Runeterra daily quests can be found here.

Repeatable Quest (3 Sun Discs)

• Rise Up
  
  • Win matchmade or AI games (matchmade counts double) (0/6)

Event Questline (5 or 6 Sun Discs)

1. Only the Worthy (5 Sun Discs)
   
   • Seize your destiny and summon Champions (0/8)
2. Claim to Glory (5 Sun Discs)
   
   • Reveal the desert's fortune and spend mana on Shuriman cards (0/150)  OR
   • Spend mana (0/270)
3. Battleborne (5 Sun Discs)
   • Use a single unit to kill multiple enemies in a match (3 or more count triple) (0/10)  OR
   • See more than 3 units die in the same round (0/25)
4. Creation's Judgement (5 Sun Discs)
   
   • Manifest victory by dealing damage with created cards (0/60)  OR
   • Deal damage (0/500)
5. Favored Fighter (5 Sun Discs)
   
   • Summon the same unit at least twice in a match (3 times or more counts triple) (0/10)  OR
   • Summon units (0/90)
6. Favored Incantation (5 Sun Discs)
   
   • Cast the same spell at least twice in a match (3 times or more counts triple) (0/12)  OR
   • Cast spells (0/70)
7. Ascended's Call (5 Sun Discs)
   
   • Answer the desert's calling and level up Shuriman Champions (0/15)  OR
   • Play Champions (0/35)
8. Future Sight (5 Sun Discs)
   
   • Shape your destiny and Predict (0/40)  OR
   • Draw cards (0/165)
9. Sacrificial Bond (5 Sun Discs)
   
   • Prove your faith and deal damage to or slay your own units (kills count triple) (0/90)  OR
   • End rounds with more units than your opponent (0/70)
10. Might of the Desert (6 Sun Discs)
    
    • Show the strength of the sands by dealing damage with Shuriman Champions (0/180)  OR
    • Attack with Champions (0/55)
11. Overkill (6 Sun Discs)
    
    • Assert dominance and overkill the enemy nexus (0/105)  OR
    • Win games with extra nexus health to spare (0/220)
12. Majesty's Reign (6 Sun Discs)
    
    • Celebrate the desert's legacy by playing cards from the Empires of the Ascended set (0/175)  OR
    • Play matchmade games (Wins count triple) (0/40)

Special Quests (4 or 6 Sun Discs)

Seems like there will be one special questline every week.

• Taliyah's Stoneweaving (1 of 2) - 4 Sun Discs
  
  • Call upon the desert's strength and summon landmarks (0/25)  OR
  • Summon units or landmarks (0/80)
• Sivir's Ricochet (2 of 2) - 6 Sun Discs
  
  • Sway the tide of battle by activating Reputation effects (0/30)  OR
  • Attack with units that have 5+ power (0/60)
• Azir's Call (1 of 3) - 4 Sun Discs
  • Grow your army by summoning units with spells and activated effects (0/20)  OR
  • Summon more than 1 unit in a single round (0/45)
• Azir's Ascension (2 of 3) - 6 Sun Discs
  • Restore what was lost and countdown the Sun Disc (0/250)  OR
  • Play rounds (0/200)
• Azir's Might (3 of 3) - 10 Sun Discs and Azir icon
  
  • Discover true power and play cards from the Emperor's Deck after leveling Azir (0/30)  OR
  • Play cards (0/300)
• Nasus' Edict (1 of 3) - 4 Sun Discs
  • Command the desert's wrath and slay enemies with Shuriman units (0/40)  OR
  • Slay enemy units (0/150)
• Nasus' Fearful Reverence (2 of 3) - 6 Sun Discs
  • Expose the enemy by striking their nexus with Fearsome units (0/70)  OR
  • Strike the enemy nexus with units (0/100)
• Nasus' Charge (3 of 3) - 10 Sun Discs and Nasus icon
  
  • Inspire the front lines and raise the power of allies in play (0/65) OR
  • Block with units that have 5+ attack or health (0/100)
• Renekton's Bloodthirst (1 of 3) - 4 Sun Discs
  • Show no mercy by dealing damage to the enemy nexus with Overwhelm units (0/120)  OR
  • Deal nexus damage (0/200)
• Renekton's Dread (2 of 3) - 6 Sun Discs
  • Cull the meek by challenging enemy units (0/70)  OR
  • Strike enemy units (0/200)
• Renekton's Fury (3 of 3) - 10 Sun Discs and Renekton icon
  
  • Channel the desert's rage and deal non-combat damage (0/130) OR
  • Deal damage to units (0/500)