Starting a Quest is a relatively easy task.
Begin by travelling to /warp Quests and right clicking the Quest Master
Once you right click the Quest Master, it will open this menu:
From this menu, select the emerald titled 'Quests' which will open this menu:
From there, select the quest category you want to complete, for this example, I will pick mining
Select the first item in this menu (in this example the wooden pickaxe) as you need to work your way through the Quest line starting at the beginning. Once you select that quest you will be met with the confirmation menu:
Click the green wool to confirm your quest! Once the quest is confirmed, you will find instructions in chat and above your hot bar: