There are plenty of reasons to go to Ishgard in Final Fantasy XIV, but one of those reasons is to visit the Firmament, an area introduced in Heavensward that serves as one of the easiest ways to level up crafters and gatherers in the game. The currency earned there, Skybuilders' Scrips, has some pretty awesome rewards.
After you've unlocked the Firmament, players will have access to several things inside it. The first is the Diadem, which allows players to gather items with their Miners, Botanists, and Fishers that they can then turn in. The main area also houses an NPC which collects crafts made specifically for the Firmament, some of which award Skybuilders' Scrips.
Getting Skybuilders' Scrips will come naturally for most players as they level up their crafters and gatherers, since part of the leveling process is running around and gathering in the Diadem, or crafting things for Ishgardian Restoration. Both of these things will reward the currency, and both will continue to be the method for gaining more Skybuilders' Scrips when you're done leveling.
How to farm Skybuilders' Scrips in FFXIV
To farm this currency, you'll need any job at least level 60, since that is when you'll get access to the Firmament. Find the Recruitment Notice in the Foundation and pick up the quest "Towards the Firmament" to get access to the area. Then, talk to Augebert once you're inside to continue and learn about all of the different areas.
After that, players have a variety of choices in how they can farm up Skybuilders' Scrips. Depending on which Disciples of Land or Disciples of Hand you have unlocked will also influence your choice. When leveling your crafters and gatherers, the Firmament and the Diadem are two obvious places where leveling will be the most fruitful.

How to farm Skybuilders' Scrips as a crafter
Because using the Firmament is the preferred method for the bulk of leveling crafters, you'll automatically earn Skybuilders' Scrips just by crafting the highest tier craft in the Ishgard Restoration menu. You can find which item to craft by heading to Potkin, the Collectable Appraiser at X 12.2 Y: 14.6. At the top will be two crafts that give a different amount of Skybuilders' Scrips.
Depending on the collectability of the item, players will gain anywhere from 66-80 scrips for the top listed craft, and 31-63 scrips for the second. The subsequent crafts will also get you scrips, but they'll be far less than the top two. If you do the green recipe, you will also gain points towards your Skyward Score, so if you're farming for scrips and also looking to increase that score, you should prioritize the one that awards the points, even if the craft might be harder. You can use tools such as the FFXIV Crafting Optimizer to build you a macro once you've collected all of the materials.

How to farm Skybuilders' Scrips as a gatherer
Speaking of collecting materials, you'll also be able to gain Skybuilders' Scrips while running around collecting in the Diadem, an instanced part of the Firmament which is accessed by Miners, Botanists, or Fishers by talking to Aurvael, the Mission Commander at X: 10.8 Y: 14. You'll queue up for an instance of the Diadem and get sent in.
From here, you have your choice of gathering with whichever job you'd like, but the best way to farm scrips is to test your patience with fishing. If you're down to fish for your scrips, head to the Swirling Cloudtop at X: 13 Y: 25. You'll be looking for the Grade 4 Artisanal Skybuilders' Little Thalaos. This fish is mooched from the Grade 4 Skybuilders' Ghost Faerie, which can be fished up using any of the Diadem baits. The Grade 4 Artisanal Skybuilders' Little Thalaos is worth 124 scrips, but you can also keep any fish that you've caught along the way, because those will give scrips as well.
As a Botanist or Miner, you'll want to run around in a circle and farm up all of the Grade 4 materials you can find. Anything lower may reward Skyward Score, but won't get you any scrips. Remember that nodes spawn in different directions depending on whether you're a Botanist or Miner, and only two nodes spawn at a time, so make sure that you gather both nodes so that two keep spawning as you run through the Diadem.
Getting Scrips from Skyrise Celebration FĂŞtes
Every three days, FĂŞtes come to the Firmament. One these days, FATEs will pop up every two hours for a 30 minute period. Scoring a gold ranking in each FATE will reward 100 Skybuilders' Scrips. If you're really gung-ho and want to pull an all-nighter, all 12 FATEs will earn you a whopping 6,000 scrips. However, even if you don't hit them all, scoring gold in all of the FATEs will get you 500 per event. Considering it's only a half hour, that's a pretty good rate of return for such a short amount of time.