Dungeon Quest Question

Sounds like you're having a tough time with the promotion mission. Check your in-game character menu or the job change tab. You can find the detailed conditions by clicking the question mark icon next to the mission, hmm!

If you look at the bottom left, you'll see Summoner > Level Up. As you level up, you'll see an option for Promotion appear.

When you level up in increments of 10, a promotion appears, but you need to meet 3 conditions. So, you should consistently do the main quests that appear on the left.

A promotion menu appears every 10 level-ups. Clear the required dungeons there to get promoted.

Summoner promotions have 3 different conditions that are assigned each time you promote. Because of that, you need to pay attention to the guide mission quests as well. Otherwise, you won't be able to open the dungeons you need to.