Now that you're officially the Champion of the Alola region, the four guardian deities will accept your challenge whenever you visit them in the deepest part of their ruins.
Ruins of Conflict
You encountered Tapu Koko, the guardian of Melemele Island during the course of the story; perhaps you even caught it at the very end. Whatever the case, you should visit the Ruins of Conflict anyway.
To reach there, head for the back of Iki Town on Melemele Island. As you approach, you'll bump into Hau. Next, go all the way up Mahalo Trail and across the newly-repaired Plank Bridge.
On the other side, grab the Electric Seed to the east. Further north is the entrance to the Ruins of Conflict. You probably saw it during the ending sequence, but there's a Zygarde Cell (14) to the right of the ruins entrance.
If you haven't caught Tapu Koko yet, head inside the ruins, then proceed north towards the deepest part. Go up the steps and, when you're ready, inspect Tapu the at the top to challenge the guardian.
The guardian deity of Melemele Island has the Electric Surge Ability, which summons an Electric Terrain at the start of battle. While this terrain is around, Electric-type moves are powered up and Pokemon cannot fall asleep (only applies for Pokemon on the ground).
Tapu Koko has good Special Attack and impressive Speed; factored in the boost from Electric Terrain and you've got a difficult battle on your hands. To make things even trickier, its signature move Nature's Madness can halve your Pokemon's current HP like Super Fang.
Ground-types are ideal since they are immune to its Electric-type moves. Of course, you should avoid hitting it with super-effective moves in case it faints. Instead, switch to a Ground-type after weakening it with False Swipe, etc.
Since Tapu Koko is an Electric-type, it cannot be paralysed. So that's one option out of the bag. In addition, Electric Terrain prevents it from falling asleep. However, Electric Terrain only lasts 5 turns, after which you're free to send it to dreamland.
Ruins of Life
This is the first ruin you visited during the story, the place where you took on Olivia during her grand trial. If you forgot, it's located just beyond Memorial Hill on Akala Island. Inside, there's a boulder puzzle to test your brain.
From the start, head north along the east side. Call Machamp and shove the east-most boulder north into the gap on the floor. After this, step on top of the sunken boulder and shove the boulder to the left into the nearby gap.
Next, move on top of the newly sunken boulder, then push the boulder above all the way north into the next gap. From here, the exit is within walking distance. As before, inspect the Tapu statue at the far end to rouse the guardian.
Tapu Lele has rock-solid Special Attack, which combined with Psychic Terrain, makes its Psychic attacks really leave a mark. It has Nature's Madness like Tapu Koko, plus Moonblast for STAB and Flatter to confuse your Pokemon.
Steel-types are the Pokemon of choice here, but try not to use their Steel attacks since Lele is kind of squishy. That way, you can resist its Psychic and Fairy attacks. Dark-types would've been nice, but they take big damage from Fairy moves. Unlike you've got Muk or something.
Ruins of Abundance
To reach here, you'll need to navigate the treacherous Haina Desert. If you haven't made it through the desert yet, flip back to the Ula'ula Island section. Inside the ruins is another boulder puzzle.
From the start, ignore the two nearby boulders and head north along the east side. Call Machamp and push the third boulder north one space, then stop (although it doesn't matter if you push it more). Now go west and push the next boulder two spaces north.
Afterwards, the exit should be within sight. Beyond, go up the steps and inspect the Tapu statue at the end to challenge the guardian of Ula'ula Island.
Like its appearance suggests, Bulu is bulky with excellent Attack and good defensive stats, but mediocre Speed and (surprisingly) HP. The one you face has Nature's Madness, but no Grass-type moves, instead Psychic, Bug and Normal-type moves.
As such, Steel-type Pokemon are good to have around to resist Tapu Bulu's heavy-hitting moves. Because it's a Grass-type, Grass-type status moves like Sleep Powder and Spore don't work, but you can put it to sleep etc. using other means.
Once you've done here, you need to exit the ruins, which can be harder than getting in... Head down the east side, ignoring the two nearby boulders. Next, shove the third boulder all the way south towards the entrance. Finally, push the boulder on the left to clear the way.
Ruins of Hope
This one couldn't be easier. You already visited the depths with Hapu towards the end of the story. In fact, you'll find the puzzle is still in its solved state, allowing you to casually stroll to the Tapu statue at end. Carefully touch it and the guardian will bless you with a battle.
Stat-wise, Fini focuses on defence and staying power. Not that its other stats are too shabby. It has Muddy Water and Hydro Pump for STAB, Nature's Madness to halve its opponent's HP and Aqua Ring to constantly restore HP.
Pokemon that resist Water-types are a boon, such as Grass-types. Since it's very likely to use Aqua Ring, you'll need to use False Swipe on it every now and then. You should also wait until Misty Terrain is gone so you can apply a status affliction.