The pipeline
Brewing is always: water bottles + Nether Wart = Awkward Potion, then a main ingredient defines the effect, then optional Gunpowder turns it into a splash potion and Redstone or Glowstone extends or amplifies it. Blaze powder fuels the stand — one powder brews 20 items.
The starter set
Sugar = Speed, Blaze Powder = Strength, Magma Cream = Fire Resistance (the single best Nether potion), Golden Carrot = Night Vision, Pufferfish = Water Breathing, Ghast Tear = Regeneration, Spider Eye = Poison. Extended versions last 8 minutes; the amplified (level II) versions are shorter but stronger — extended beats amplified for exploration.
Sourcing in bulk
Nether Wart grows on soul sand in the Nether and in any dimension — bring a starter batch home and farm it beside your brewing stand. Blaze rods from blazes near the fortress cover fuel forever. Water bottles need nothing but glass and a cauldron or water source.
Potions that change fights
Strength II splash potions trivially speed up boss fights; Fire Resistance makes the Nether breathable; Slow Falling (Phantom Membrane) turns Elytra experiments from fatal to cheap. Carry splash versions so you can buff mid-combat without opening the hotbar menu.