Here is a video of a quest from final fantasy 14 and how you interact with one. You can see you first select from a list of different available quests and you look at the quests requirements and rewards. From there you select the quest and then decide what level of difficulty you want to do because you can select different challenges and include more party members to help you. The quests are also rated in a star form from hardest to easiest.
When you accept the quest then a quick little action is performed and the quests icon is shown on your character. When you are ready for the quest to being you travel to the select area for the quest. Then y u begin doing the quests requirement such as the one shown in the video has you killing multiple mobs. Each time you kill one you get a quick update on completing that part of your quest, but it may differ based on what quest you are participating in completing in the game.
When you have actually completed the quest you will get a little rift or status beacon shown. You then can interact with it. You just have to click on it and then select the quest panel that is shown in the upper right hand side of the selected object. From there the quest log will pop up and you can not only have the quest completed but you get instantly all the perks that are associated with completing the ffxiv quest. Best of all unlike other games you can now get all your rewards right away because you will be given a way to teleport to a variety of different areas in the game. This is one of the best parts of the new questing system because its much more streamlined than before. In the old final fantasy 11 you would have to run all the back to complete the quest but now you can avoid
Anyways thats a quick little beginners questing guide to final fantasy 14. Enjoy.