Each day, you are able to take delivery orders from 3 different NPCs. Completing each delivery gives you some rewards, and completing all 3 for the day will reward you with 60 Primogems. Go to the event page and tap the accept button to navigate to the NPC to start the quest.
Each dish has its own unique restriction for the delivery. In the example shown, you will have to deliver the food within the time limit while also not glide, sprint, or take any damage. The delivery will immediately fail if you do any of the restricted actions.
Since it won't show you what the restrictions are once you accept the quest, if you forget what they are and want to check again, open up the event page (F5 on PC) by tapping on the event icon in the upper right part of the screen, left of the Battle Pass icon.
The timer will start immediately after you talk to the NPC to accept the quest. You are able to teleport around the map freely. However, do keep in mind that the timer will keep going even when you are opening up the map. It's definitely advantageous if you've unlocked the teleport waypoints all over the place for an easier time.
If you fail the delivery, you can retry by talking to the NPC that gives you the food again, and the delivery destination will be the same. This means that you can strategize your delivery if you are having trouble with it.
If you are really struggling with a particular delivery quest, you can consider using the portable waypoint item. You get the blueprint to craft the portable waypoint by reaching level 6 in your Mondstadt's city reputation. Use this item close the person that you have to deliver it to, accept the delivery quest and simply teleport to the teleport waypoint for a really easy and fast clear.
You can only have 1 portable waypoint available in your map at a time, placing another one will remove the old one. Each portable waypoint only lasts for 1 week. If you want to farm, or find some fireflies, click the link below for some easy and known places to get them.
You can reduce your stamina cost by using a "travel party":
By using 2 anemo element characters, you will get the elemental resonance that reduces stamina consumption by 15%, as well as +10% movement speed and -5% skill cooldown.
Kaeya has a passive talent that reduces sprinting stamina by 20%. If you are not allowed to sprint but can swim/glide, then switch to an appropriate character that have passives to help with those actions like Beidou for swimming, or Venti/Amber for glide.
Barbara and Diona's elemental skill has an effect that reduces stamina consumption. Just remember not to use this if the restriction says you can't be affected affected by an element, because their elemental skill applies an element on your character.
There are also plenty of food that reduces stamina consumption. When you want to cook some foods, use the filter at the bottom left side of the screen and set it to "Adventurer's Dishes." Most of them reduce sprinting stamina consumption, but some can reduce climbing or gliding.