Brew a Potion of Swiftness by placing an Awkward Potion in a Brewing Stand with Sugar
Add Gunpowder to the Potion of Swiftness to make a Splash Potion of Swiftness
To make Speed II, first brew a Potion of Swiftness II by adding Glowstone Dust to the Potion of Swiftness
Add Gunpowder to the Potion of Swiftness II to make a Splash Potion of Swiftness II
Alternatively, add Gunpowder first to make a Splash Potion of Swiftness, then add Glowstone Dust to upgrade it to Speed II if your version allows it
Use Blaze Powder as fuel in the Brewing Stand
Start with Water Bottles and Nether Wart to make Awkward Potions before brewing Swiftness
Use Redstone only if you want a longer duration, not Speed II