How do you actually do the missions? I do not understand them.

Guide to Mission Pets and Missions

basically, you go to the barracks, and it'll have 5 missions (I think higher levels are gated behind town reputation but I Cannot Remember because it's been nearly a decade since that was relevant to me)
each level difficulty has five potential missions it can offer- one for each of HP, str, int, dex, and agi- selected to offer randomly each hour.

If you click on "more Info" it'll bring up a popup, and that will have the following kind of info:

Mission Name: Planting Seeds
Required Level: 0
Difficulty: Easy
Emphasized Statistic: Intelligence
Duration: 01:00:00


Mission name is always the same for the level+stat combo, eg. Planting Seeds is always the lv0/easy Int mission
Required level means the pet has to be at least that level to go on that mission, a lv1 cannot be sent on a lv10 mission etc. The Level of the mission can be used as a general guideline of about where you expect how much a pet needs in a stat to generally be pulling Satisfactory (the average result) with available prof points in the Emphasized Statistic.
The Emphasized Statistic is the stat used in the rng dice roll that determines how you did. It takes the Active stat into account for this (the first number in a stat line eg, the 449 in "Dexterity: 449 (78 + 0)" (the number that's the base stats+proficiency bonus+level bonus+equip bonus)

so you want to send a pet that's been proffed and equipped for the stat the mission is asking for.

Duration is the length of time the mission will be going for. You cannot use a pet for other things while it's on a mission, and while you have a pet on a particular mission, you cannot send another on that same mission (you can only have one Planting Seeds at a time, for example, so if the hourly mission roll picks Planting Seeds as one of the options while a pet is on that mission, the 'more info' button will be greyed out and unclickable, but will become clickable per usual if you end or complete the mission on the Mission Log page)

You select the pet you want to send on the mission from the dropdowns, hit accept mission, and then once the timer of the mission has elapsed you can go to the Mission Log page and Complete the mission.

There are four possible outcomes to a mission, which your pet's stat will affect the chances of, along with a little bit of rng (so it's a bit like a gradient)
Failure (no reward)
Poor (half base reward+no mission satchel reward)
Satisfactory (base reward+one satchel)
Exceptional (double base reward+two satchels)

Thank you very much. I was confused.