Whatever helps with immersion, which is a little bit of everything, though mostly the story and its characters, I feel. I don't care much about the music or graphics/art, even in games, anime or manga.
Da Capo II and Eien no Aselia. I didn't really like the Da Capo II anime, but the drama and comedy in the visual made me rethink my opinion of the Da Capo franchise in general; in fact, now I'm a Da Capo fanatic. Eien no Aselia was a great SRPG, which I enjoy playing, and the visual novel element made it even better because it offered an immersing story. Sharin no Kuni is an honorable mention.
I'm not a completionist, so I'll just do the routes I enjoy, unless I really loved the visual novel and want to get 100%. For me, there's no point in getting 100% in anything I did not like.
I was really unsatisfied with whom Rin had chosen in the end of the Shuffle! anime, so I wanted to play as him and choose the "right" heroine. Next I moved onto Da Capo and I became hooked on them.