Advices
You get one free portal charge everyday, but only if you have less than three
You get another one from the last prestige reward
It is advised to leave your completed adventures hanging for the next day, so that you always start your day by raiding. This way you can still complete one daily adventure if things go wrong
It is very important to reach the higher-level adventures as fast as possible, since an adventure 13 reward about 200 chaos particles a day
Closing your adventure does not close it for other players
If you enter an adventure where the boss has been killed, you will immediately be presented with the rewards, however you can still close this screen and fight the remaining spots
Enemies benefit from random daily buffs, you can see them in the screen where you create an adventure. They can significantly alter the difficulty
Higher-level adventures have spots with resistances against some damage types, up to 100% in adventure 13. This affects what characters you should raise, see the section below
If you struggle in adventures, consider a blitz team, such as one with Dante in front with Cain
Adventure 13

It is a stronger version of maps 9-12: a13-9, a13-10, a13-11, and a13-12.
The daily map is chosen randomly, and it common to all players. You can see the map of the day by looking at the picture of the boss on the portal (image on the right)
You usually want to avoid the 13-10 map, so make sure to have the previous day's adventure still hanging so that you can avoid it
You will need to be able to deal at least two damage types because of complete immunities to specific damage types, visible among the buffs list as
A good team for that is Dante+Orion
Characters that apply buffs or additional damages will not if their primary damage type is ignored: Augustus, Seb, Keira, Galahad, Yasmine, Iris, ...
Corrupted Galahad

A difficult boss to deal with because of his pure damage and vampirism. Some approaches:
Dante can stop his ult by knocking him back right as starts growing before his ult. However you will need to manually conduct the fight and be reactive
Galahad may also stop his ult with ascended skills, I assume, although it probably only slows him down
Jhu puts a decent solo against him thanks to his periodic immortality. He benefits nicely from Thea - even a weak and dead one as long as her fourth talent is maxed. Dorian can be helpful or harmful, because while he will heal Jhu, Galahad will then heal himself in return by damaging Jhu
Heidi can dodge pure damage
You can make Yasmine ult manually whenever Galahad ults himself, to get behind him. She may also be able to do a decent job in auto by sometimes causing Galahad to launch his swords behind himself
Corrupted Orion

Another difficult boss who deals a lot of damage. The good news is that he deals magic damage, so there are workarounds.
You could simply overpower him, this is not that hard
You could bring a protection against magic damage: Rufus especially, or a strong Isaac. Or, to a lesser extent, maybe Aurora or Thea (silence)
Jhu has a decent chance to solo him, see the strategy above against Galahad
Corrupted Cleaver

Provided you bring a good enough healer and damage dealers, it should be pretty trivial. Dorian is a very cheap solution to get enough healing. This boss is routinely defeated with a 200% damage buff in adventures 12 and 13-12, so he is very manageable.
Corrupted Ginger

The easiest boss, piece of cake. If you still struggle though, remember her ult deals physical damage, so bring Astaroth or characters with a high armor (not mages).