Firstly, make sure that your Foap account is verified with PayPal.
Secondly, Missions are for users who are at least 18 years old, so if you can't see Missions and you are at least 18 years old, then you would need to change your birth date in the app.
To join a Mission:
Once in the app click on the little bell icon located on the bottom right of the screen. There you will see "Current", "Joined" and "Winners" tab. Under the "Current" you can see the list with all the available Missions and the info about them under the each Mission (on the left - amount of photos uploaded, middle - prize, right - days left).
To join the Mission click on the one you want to join and then click "Join Mission" (on iOS) or little bell icon located in the middle bottom (on Android) and you can choose a photo you'd like to upload.
Once you've uploaded the photo, go to "Manage photos" in the main menu of the app and open the photo you uploaded to the Mission to add the information required (add caption or description, add at least 5 tags and answer legal questions.
Once uploaded add all of the information required (captions, at least 5 tags and answer legal questions).
Once you've completed all of the steps above your photo is submitted to the Mission and waiting for curation process to finish.
Major part of the Missions are for approved photos only, so even if you see a photo with a Mission name in your profile, it doesn't necessarily mean that they have been approved to the Mission. It can also mean that either they still haven't been curated or they haven't been approved.
If your photo appears among the Mission photos (under the "Submitted" section) it means that it has been approved in the Mission and has a chance to win it.
Keep in mind that sometimes the curation process can take longer than 72 hours and it's usually fully finished after the Mission has ended (meaning the winner won't be chosen before all of the photos have been reviewed).