KRIPTINI'S GUIDE TO BEING HEAVENSWARD-READY AS FAST AS POSSIBLE:
1. Reach level 50 and complete your level 50 job quest and the quest, "The Ultimate Weapon," which is the final quest in the first MSQ quest chain.
2. Complete the quest, "The Weaponsmith of Legend." (This unlocks your relic quest.)
3. Complete the quest, "A Relic Reborn." (This gives you a cheap i80 weapon and earns you quite a bit of Tomestones of Soldiery.)
4. Purchase three Thavnarian Mist in Revenant's Toll for 150 Tomestones of Soldier each, then use them at the Furnace near Gerolt in the North Shroud to upgrade your relic to Zenith. (It will make your weapon i90, which is the minimum required iLevel for accessing Heavensward. If you don't have enough Soldiery at this point, skip this step and come back to it later.)
5. Comeplete the quests, "Legacy of Allag," "Sanding it Down," "A Performance for the Ages," and "Labyrinth of the Ancients." (This unlocks the Labyrinth of the Ancients raid.)
6. Farm the Labyrinth of the Ancients raid. Dropped loot will be i80 left side pieces. Exchange any Tomestones of Soldiery you get for i100 right side (jewelry) gear in Mor Dhona. Keep doing this until you have an average item level of i70. (You need an average item level of i50 in order to enter LotA. If you have your Zenith and quested gear, you should be okay.)
7. Once you have an average item level of i70, complete the quests "For Prosperity" and "Syrcus Tower."
8. Farm the Syrcus Tower raid. Dropped loot will be i100 left side pieces, as well as "Oils of Time" which you can use to upgrade your i100 jewelry to i110 by talking to Gerolt's apprentice, Drake. As before, exchange any Soldiery you get for i100 right side jewelry in Mor Dhona if you haven't finished gearing that side by now. Repeat this until your average item level is i90.
9. Complete the second MSQ quest chain, starting with "The Price of Principles." (This will take a very long time, and the last few quests can't be done unless you are at least i90. It's not a bad idea to work on these while you wait for raid DF queues.)
Once you finish step 9, you're ready for Heavensward!