This page of the Disgaea RPG guide answers where to find and how to put your equipment.
Armor and weapons can be found in the Rosen Queen, in the Shop tab. You will buy them with HL coins. Keep refreshing the inventory list until you find rare or legendary items.
Once you have purchased something, click on the Party icon located at the bottom of the screen. Go to the Character List, and then select the character you want to use the new item.
Then click on Equip. The Weapon window will open. Select one of the available weapons and click on the Equip icon at the bottom of the screen. Try to always equip a character with their preferred weapon type. The recommended weapon type icon can be found on the hero's thumbnail.
When changing equipment, you can check the statistics of the current (Equipped) and Selected weapon.
If you do not have time to browse through all the items so you can choose the best one, click Auto. The game will then automatically assign the most powerful item to your hero.
Besides weapons, your character can wear a maximum of 3 armor sets. Go to Armor 1, 2, and 3 tabs and select the best items.