In addition to forging your equipment, you can also upgrade your equipment. Upgrading armor will increase its defense (deviant equipment has a special upgrade system and so has its own section, "Deviant Equipment"), and upgrading weapons will increase its attack power and sharpness, and also elemental power and affinity (or it could decrease a weapon's affinity!).
Upgrading in this game has been overhauled and has many added features that make it a little more complex than in previous installments. There are two distinct parts to upgrading: Level of gear, and upgrade branch of the gear. Note that armor does not have an upgrade tree, only levels.
Gear can level up. Armor is by far the simplest to upgrade so we'll start with that. Armor, when upgraded, will have increased Defense, which can be exceedingly helpful in getting that last bit of Defense you need (particularly in High Rank since aside from upgrading armor that's the highest the Defense gets). Armor requires three things to upgrade: zenny, material points, and armor spheres .
Zenny is pretty straightforward, and material points are a new mechanic in this game. Monster parts are all worth a specific number of monster points ranging from 1 - 3 points. Common materials are worth 1 point, but rarer materials are worth 3. Armor will require you contribute materials until you hit the cap (or accidentally overshoot over). Then you pay up zenny, and use the next component.
Armor spheres are a special item that can be fused into gear to fortify it. They can be mined and the lower grade armor spheres can be purchased from Neko (means "Cat"). Armor sphere grades are indicated by their name (for example, Advanced Armor Spheres are better than Armor Spheres+ are better than Armor Spheres ). As you upgrade through the levels, you will need higher grade armor spheres.
Weapons, new to this game, can also level up. Unlike armor, they don't need "weapon spheres" or anything of the sort, but rather require zenny, monster parts, and miscellaneous ores or bugs. Different weapons have different level caps. Leveling up a weapon will pretty dramatically increase its power and sharpness, so it's more visibly immediate when you have upgraded weapon than when you have upgraded armor. This also results in almost every weapon being viable in the endgame because every weapon in an upgrade tree has a level cap.
A weapon's level is kept independent from where it is on the upgrade tree. As you level up a weapon, you will unlock different branch possibilities. You can select to move onto a branch, or just keep leveling up a wepaon. Note than when you move onto a upgrade branch, the level is reset back to 1 (the materials needed to upgrade will change, however).
Upgrading a weapon will change what the weapon looks like and which monster or general theme it is based off of. For example, if we max level a pair of Petrified Daggers, we have the following upgrade options: Insecticutters, Type 51 Macerators, Twin Chainsaws, Gelid Mind, and Obsidian Daggers. At max level, Petrified Daggers are perfectly usable. You could never touch a single upgrade, just level up the starting weapon you have, and it will be fine to use!
However, if you want to spice things up a bit, you can select one of those upgrade options and get a much cooler-looking weapon. This resets the weapon's level to 1. Changing the upgrade path might cause a slight downgrade in power, but as you level the weapon up, it will be quickly compensated for.
Note: Not all weapons are locked within upgrade trees. Many weapons are fairly far advanced into the upgrade tree but can be forged immediately if you are willing to part with more zenny and more monster parts. This lets you conserve resources in the long haul by not having to make a ton of starter weapons and upgrade them all differently. Be constantly checking what's available at the Smithy before you commit. Other weapons, unfortunately, are gated within the upgrade tree, which means you're out of luck there.