First, join or create a team by clicking
My Team button in inventory, top-left corner. You can skip this step on singleplayer and most new modpacks, as you already are in a team by default.
After that, click on the Quests button, and a GUI will open:
You can change chapter by either clicking on the left side tab or pressing Tab key or 1-9 keys.
Click on a quest to see required tasks to complete it and rewards you can get for it. Some quests can require you to complete other quests before you can view it.
Click on task button to submit items from your inventory if the task consumes items, otherwise it will auto-detect them.
Click on reward button to claim a reward. If the reward name is blue, then it means only one person can claim the reward (It's called Team Reward because the whole team gets one reward (I know, the name is pretty unclear, I will figure out something better)), otherwise, everyone on team will get a copy (default).
Some tasks can't be completed from the GUI alone (e.g. FE / RF or IC2 energy tasks), so you may need to use a Task Screen block. More on task screens here.