Your first objective will be to level your Engineering skill to at least level 200. There is a Gnome-centric Engineering guide that you can follow here. The only thing to keep in mind that will be different from the guide if you follow it, is that you will want to put mats towards making one of each new type of goggle as it becomes available.
The first goggles you can make are the Flying Tiger Goggles at 100 skill points in Engineering. You will need a pair of these to later create Green Tinted Goggles at 150 skill points. You will then turn the Green Tinted Goggles into Fire Goggles at 205 skill points.
The most accessible trainer, Tinkmaster Overspark, is in Ironforge. However, in order to complete the quest, you will need an Accurate Scope, which is not trainable by a regular trainer but is only available from Mazk Snipeshot, who hangs out in Booty Bay, along with another Gnomish Engineering Trainer, Oglethorpe Obnoticus. So if you have to go down to Booty Bay to pick up the schematic, it’s also possible to also pick up and turn in Show Your Work there.
Accurate scopes (as well as the rest of the required objects) are not BoP, so it’s possible your guild bank might already have some, or that another Lilli is around to craft what you need.
Once you have turned in the quest, you are able to speak to any of the Gnomish Engineering Trainer to get those schematics only available to Gnomish Engineers. Now the only thing that stands between you and your Gnomish Goggles is getting to 215 skill points in Engineering and training the schematic from one of the Gnomish Engineering Trainers.
So, to recap:
- Step 1. Level Engineering to 200, being sure to make all possible goggles along the way
- Step 2. Pick up Show Your Work
- Step 3. Craft items needed for the quest to continue leveling Engineering, as you are able
- Step 4. Turn in quest
- Step 5. Level Engineering to 215 and learn Gnomish Goggles schematic from a Gnomish Engineering Trainer
- Step 6. ???
- Step 7. Profit