For pre-hardmode tools:
Increase Health: Life Crystals, discover them by exploring in caves.
Increase mana: Mana Crystals, collect stars at night
Breathing in water: Diving helmet, or gills potion, kill a lot of sharks at the ocean, it may eventually drop, gills potion is easier to craft, but luck based.
Rising quickly to the surface of water: Inner Tube or flippers, both can be found at underwater chests.
Moving through lava: Lava Shark Mount, can be obtained by fishing in lava at the earth's bottom layer, but it needs hot critters to work as bait, which in turn needs a hot bug net to be made. It's a pretty rare possibility to move quick around lava. Obsidian Skin Potions or a Lava Charm works too to be temporary immume to lava.
Illuminate dark places: Mining Helmet, can be bought at a merchant. Placing down torches will work too