Upgrading in Final Fantasy VII Remake rewards SP, which is used to upgrade and upgrade weapons.
Each weapon in Final Fantasy VII Remake comes with a full-fledged skill tree (which reminds me a bit of the grid of spheres in Final Fantasy X, but much simpler). After speaking with Biggs and Wedge in the Sector 7 slums at the start of Chapter 2, you will unlock the ability to upgrade your weapons, and a new option will be added to the main menu.
In the Upgrade Weapons submenu of the main menu, you can see what the levels of your weapons are and how much SP they need to spend. SP is earned by leveling up your characters, and it goes into a general pool for each weapon (even unused weapons will receive the assigned SP). At certain SP thresholds, your weapons will increase, which opens up another core on the skill tree.
Each core contains a handful of stat bonuses tailored to that weapon's playstyle. Tifa's Sonic Strikers, for example, lean towards casting magic, and the second core offers a wealth of + 5% damage increases to various magic elements.
Boosts for each core will cost more and more SP to unlock, from four, six, eight, etc. There are five levels per weapon, so five hearts. The UI in the core upgrade menu will display at the top right the number of cores you have, the number of boosts you have unlocked in that core, and the SP cost for the boosts for each core. It's a pretty informative user interface.
SP spent on one weapon isn't spent on another, so each can be upgraded independently without worrying about saving SP for either. Simply earn SP as you level up and upgrade your weapons as you see fit, even ones you aren't actively using.
If that is too exhilarating for you, you can automate the process from the upgrade menu. Press Triangle to choose between Balanced, Attack, and Defense, and the game will automatically assign SPs to boosts that match your selection. Tapping Square will force the game to prioritize your equipped gear over everything else.
I prefer to refine my weapons myself, but if you just want to experience the story without the micromanaging, automation is a great way to do it.
If at any point you want to reset your weapons and reassign SPs, you can do so by talking to Chadley (the guy who provides you with new materials to complete the Battle Log challenges). It will cost you gil, but any SP spent on this weapon will be refunded.
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