Shirou is the main character of the show, so he has relationships with almost all the characters. Archer appears for a very short time compared to him, so although I can make comparisons Archer’s interactions would result as fewer or less detailed. I may also say that most of these pages are based on the game only and that I take Unlimited Blade Works and Heavens Feel as the most significant scenarios (so that's why I state that Sakura and Shirou are a pairing, and there's no striking love interest between Shirou & Saber). I’ll skip Hollow Ataraxia characters since the Shirou featured there isn’t the real one (but Avenger disguised as him) and Archer hasn’t that big part either.
Just click on the name of the character to get to the explicative page.

emiya shirou: face to face with himself.
tohsaka rin: friend and master.
saber: the servant, a rival.
matou sakura: love and despair.
ilya: sister and enemy.
lancer: friend and rival.
fujimura taiga: sister and burden.
matou shinji: traitor and enemy.
emiya kiritsugu: father and redemption.
kotomine kirei: true evil and fool.
gilgamesh: rival and fool.
rider: ally and enemy.
ryudo issei.
kuzuki souichirou.

More coming soon!