This legendary favor will immediately start after killing the first valkyrie. Since most valkyries are hidden inside magically-sealed Hidden Chambers of Odin, the earliest you can attempt this favor is after completing the journey quest Chapter 7: The Magic Chisel where you’ll get the titular chisel to break the seals to these hidden chambers. The easiest valkyrie that will serve as a good Introduction to valkyrie combat is Gunnr, which is locked in the hidden chamber by the dock in Thamur’s Corpse. You can visit her chamber as soon as you get back from retrieving the chisel.
After defeating a valkyrie, she will leave behind her helmet which is also part of another legendary favor (The High Council), Asgardian steel (used for upgrading valkyrie gear), enchantments, Valkyrie armor pieces, pommels, etc. Stronger valkyries will leave behind epic drops upon defeat as well.
We have dedicated pages for the individual strategies for each valkyrie as listed below. Aside from Gunner being the easiest of the 8 valkyries, you can challenge them in any order you want. There are convenient mystic gateways in each chamber (except Hildr’s since she’s hidden in a randomized part of a maze) so you can always retreat and return when you have better gears and skills.