The anime adaptation of F/SN mostly focussed on Saber's path with some dabblings of the other two paths. Various facts were alluded to which may be hard to pick up without prior knowledge. Sakura is Rin's blood sister but Sorcerer society declares that only one child per clan in each generation can be trained in the mage arts. Rin was chosen to be the successor of the Tohsaka clan, but Sakura was plenty potent too. The Matou clan's bloodline was growing dry and so Sakura was adopted by them to become the new heir. This did not sit well with Shinji, who had his place as heir stolen by someone from outside the family. To vent his frustrations, he raped her. When the 5th Holy Grail war started, Sakura summoned Rider but her command spells and rights to be a Master were passed onto Shinji. While this history is not directly shown in the anime, this is why Shinji's command spells are inscribed on a book and not his hand. The astute viewer would note that there is tension between Sakura and Rin when they first appear in the same scene, and that the normally formal and polite Sakura drops to a casual tone for Rin.
Sakura fell in love with Shirou back when he was a child. The sight of a young kid trying his hardest to do pull-ups at the park touched her heart. It's part of her daily routine to visit Shirou in the morning before school - senpai is perfectly capable of doing housework but it brings her happiness to cook for him (and sometimes he falls asleep in the garage tinkering with stuff so it's convenient for him).
In the anime, there was a brief arc where Caster took control of Sakura and planned on using her as a vessel to summon the Holy Grail without winning the war. It ended with a fight between a controlled Sakura and Rin. This is all the anime did to refer to what really happens on Sakura path. To learn of Dark Sakura, her true feelings towards Rin, the fate of rapist Shinji, the creepy head of the Matou clan and how the Holy Grail war was properly concluded (Saber path was about delaying it to the next generation, with nothing resolved), play the game.