Hello,
I created my account today and I'm struggling with missions. I entered my only pet into a mission (So now I cant do anything, great) and it was only supposed to be an hour long. (I'm pretty sure its been an hour now though and I still cant use my pet?) when I visit the missions page the "more info" button is greyed out but the timer is still at 1:00:00. Is it supposed to count down? Is there anywhere to see the time until it's finished?

Go to explore > missions. That will bring you to your mission log, where you can see all the missions you have, when they end, and cancel/complete them.

Ah thank you, very helpful!

To go further on what Wonderfulb2uty said: the Missions Page, where you accept the missions, does not tell you anything about the missions that you are currently on, beyond greying out a mission if it matches one already in progress in the Mission Log.

What the Mission Barracks display is the Mission Name, minimum reccomended level, 'difficulty level', how long the mission will take, and the More Info.
When a mission is complete, the menu 'Explore' and it's 'Mission Log' tab should be orange the next time you load a sylestia page, to alert you.

The particular missions available change every hour (at random, I believe) between the five mission types available, and if you take a mission, the 'slot' that it was in is 'taken' until next hour (when it randomizes the next lot) (Each Barrack has five 'slots' once fully unlocked- two easy, one medium, one hard, one very hard)
I'd suppose that the 'more info' was greyed out because it was the same mission that your pet was already on, or it was still the same hour (ie. if you checked at, say, 1:05pm and then then 1:56pm.)

We answered a question about possible results for Missions here, where we went a bit more in-depth with mission 'types', not sure if it'll help, but might as well offer it.