It will likely take a while to get this achievement, due to the player getting only one Epic Quest at a time from the Quest Board. If the player focuses on completing the Epic Quest (for example, playing as a fire variant when needing to vanquish enemies) the player will eventually complete the Epic Quest.
For "Get 500 Vanquishes with the Primary Weapon of any x Variant" quests, playing a couple of rounds of either Garden or Graveyard Ops can help, as there will be many enemies spawned. This also applies to the Flag of Power mode.
For "Win x games of y" quests, playing the required mode on Solo Ops or with friends are good options. It also helps to get practice on the mode you're trying to win.