Gems and the Gem Frame
Gems have the power to alter the escense of living things, and their power can be harnessed by a Gem Frame that is given to you by Watts at the beginning of the game.
This frame can be accessed via the “Gems” option in the characters menu. It will show up as a grid in which stones can be placed to make their effects active on your character. The effects of each stone vary, and can both increase your stats and give you additional abilities.
The Gem Frame starts off as a 2×2 grid, and can be upgraded 3 times during the game. These are pointed out during the walkthrough, however for quick reference here are the points where you upgrade your Frame.
1st Upgrade (4×2)
Talk to Balbo of the Bomb Brothers Store after completing Star Lake of Topple.
2nd Upgrade (cross formation, 2,4,4,2)
Talk to Balbo after finishing Brightwood of Wendal.
3rd Upgrade (4×4)
After completing the Path of Life, Chat to the Dudbear at Dud Services to access the subquest that will reward you with the Smelting Rock. Give this to the Bomb Brothers so they can make the final expansion.
7 Classes of Gems and Fusion
Click on each gem to see a full list of the gems for that colour and how to obtain them, including details of the gem fusions for that colour.
Regarding recommended gems and gem combinations, it comes a lot down to the situation you are going to enter. A few that I would recommend obtaining and learning when to use are gems such as Sword Expertise, Heart of Progress, Virtue of Balance, Ultimate Protection (this one is particularily useful for Path of Life and Cosmic Rift, as the minor obstacles can inflict random status ailments) and Soul of the Gemma.
Blue gems are not particularily useful to me, as I have found magic to be hardly an asset, so you really want to focus more on using those extra slots to add more ATK or DEF, and maybe even boost the sword specifically a bit more.
Another thing to consider is a Green gem that protects you from the status ailments that minor obstacles can inflict (for example, poisoning in Brightwood). It will come in handy for levels that require you to search through the minor obstacles for the Gleamdrop.
Also the effects of EXP boosting gems (such as Heart of Progress and Courage to Learn) accumulate side by side, so if you have two Heart of Progress, you double your EXP, as opposed to them accumulating exponentially, which would give you (100 * 1.5) * 1.5 = 225%
I recommend reading the walkthrough section at the beginning of each level for an idea of what terrain you will be facing, and hopefully this will allow you to prepare better gem combinations for each level as required.