Upgraded poetic gear (ilvl 130) will carry you through the bulk of Heavensward leveling. Around 56-57 you'll start seeing upgrades from vendors/dungeons/quests. At 60 you can nab some ilvl 200 gear (including weapons) from Centurio Seals that you get from doing the hunts given in Ishgard. That gear can be upgraded to 210 as well. Once you hit average ilvl 175 you'll have access to Void Ark which also drops ilvl 200 gear and can be run as much as you want. Running dungeons will also get you Tomestones of Lore which can buy 230 gear.

Basically, once you hit ilvl 175+, getting gear upgrades shouldn't be too difficult.