Congratulations on hitting level 50!

* For weapon, I recommend starting with either Flame Lieutenant's Bow (bought with GC seals) or Darklight Composite Bow (drops from Amdabor Keep and Wanderer's Palace). Longarm Composite Bow is the 3rd option, and as KinnisonArc said above, you will need one anyway for your relic quest. It's crafted and can be bought from the Market Boards (although you should check if someone in your FC could craft one for you cheaper if you give them the materials).

* In addition, you should unlock your relic weapon quest to start working on your Artemis Bow. This page has all the instructions you need.

* Most of the new dungeons unlock from quests starting from either Vesper Bay or Revenant's Toll. Check this page and the list of quests unlocked after completing main scenario to catch them all.

* For gear, you should aim for getting as many item level 90 mythology pieces and item level 100 soldiery pieces as possible. You can farm 450 soldiery every week (resets on Tuesday) and unlimited amount of Mythology. You get soldiery from so called expert dungeons (longstop hard, amdapor city and halatali hard), and you may want to gear up a bit before challenging those. Mythology is rewarded from all level 50 dungeons. You also get extra mythology / soldiery rewards from running your daily duty finder roulettes.

In short, you should start working on your relic weapon quest and gearing up with mythology (and soldiery, a bit later). After finishing your relic quest you'll get access to Coil 1 and the extreme primals. After finishing Coil 1 you get access to Coil 2.