Your main method for operating your Servants and generally playing GG2 is through the Organ Menu. Getting used to it, and Overture in general, is a test of patience and strategy. Some characters, like Dr. Paradigm or Valentine, place much heavier emphasis on the Organ Menu than ones like Sin or Sol, and the more competent your Master is, the less you'll utilize this mechanic. This doesn't mean that it isn't integral to the flow of the game, but just a reminder that
The primary use of the Organ Menu is commanding Servants. While in the menu, you can hit X to Summon Servants, and use Mana to do so. There is a limit of 40 Servants in 1v1 battles (20 to both sides), and 32 in 2v2 battles (16 to each Master). Making good use of the Servants you're allotted is important, and if you want to get rid of some, you can delete them by hovering over them in the Organ Menu and hitting Y (do so sparingly, as your Mana is not refunded).
The goal of a match is to destroy the enemy's Master Ghost, but this can't be done without capturing the smaller points, known as Ghosts, that are scattered around the map. They're all numbered (as shown in the Organ Menu in the Servants section). Capturing one prevents enemy Capture UnitsSmall enemies that inflict massive capture progress on Ghosts. They are automatically produced by Master Ghosts and Ghosts that have been captured. from being sent and increases your Mana Gain. It's recommended to have at least some units sitting on a Ghost if it's tactically important to you, as they'll be taken by random enemy Capture Units otherwise.
These are purchaseable abilities that each Master has access to. While some, like Sin's Humming Sword, aren't completely essential to that character, others, like Ky's Charged Stun Edge or Sol's Dragon Install, are pivotal. They all cost Mana, and most remain for the rest of the round/match (though some are single use).
Items are important to the versatility of your Master. While they aren't always needed to turn the tide, ones like Resurrection and Life Potion (L) can change a battle from desperate to one-sided. In the Item menu, that shop depends on the time of day the stage is supposed to be, while Item2 slowly fills up over time with random items based on rarity (Resurrection and Elixir being rarest, Mouth Zipper and other cheap items being common).