Garment Grids are the "plates" that you must enter a Dress Sphere into in order to use it in battle. There are many Garment Grids available in the game, and most people do not use them to their advantage. Most Garment Grids have either an increased ability when you move from one dress sphere to another during battle through a "gate". Gates (represented by colored symbols, the legend of which you can find in the bottom corner of the screen) can do many different things like increase your MP or HP, or sometimes it gives you an additional power, such as darktouch or sleeptouch, among many others!
Each grid is different, and each grid has it's strengths. Use them wisely and fit them to the circumstances of the battles you are facing. Fighting alot of poison attack fiends? Use the garment grid with Poisonproof on it! The Grids are there to give you an advantage against your enemies, so use it!
Each grid has "nodes" or slots to enter your dress spheres into, so make sure you think about your upcoming battles when choosing the Garmet Grid and the Dresspheres in it!
Shinra's CommSphere Network
CommSpheres are very important to gaining more completion percentage, and to furthering some of the stories you are following around Spira.
It is my recommendation, that you go to every location during Chapter 3 to make sure that Shinra is able to install the CommSpheres in place then, because you will need them come Chapter 5!
There is an in depth FAQ by Jonathan Osland to be found here. This FAQ will explain every scene that you should see on your CommSpheres, and how to make sure you see everything you should.
I will warn you that watching CommSpheres can get a bit boring, and it does feel tedious from time to time, but it is worth it for the percentage completed, whether you are working on getting 100% or just enough to get the "good ending". (See Endings Section)
Shinra is the wunderkind Al Bhed boy aboard the Celcius that is the one that can turn quality spheres into Dress Spheres. If you speak to him while aboard the airship, he also offers you a glimpse into his bestiary list, which will give you a run down of all the fiends you have fought during your adventure.