Board Thread:Lore Discussion/@comment-24098614-20150903093108/@comment-6881629-20150903101028

You can pretty do both if your careful once you find Sky Haven stop doing main story quests, you only get made to make the choice between them after the quest Alduin's Bane. With the Blades make sure you do the Rebuilding the Blades and Dragon Research quest after that they only have the repeatable quest Dragon Hunting (kill a random dragon). The Greybeards give the quest Find the Word of Power to help you track down word wall and can boost the power of one of three shouts (you can keep switch between them unless you pick the Blades).

As for the main reason they make you choose between them it's a spoiler (click if you really really want to know).

Which is actually better when you do finally have to choose is up to you and your character in game. I've never sided with the Blades though, Delphine is just so incredibly stubborn.