From your home you explore the forest until you find the lake or a handsome man walking invites you to go to the lake, say yes. From then on, you can visit the lake as a location. When at the lake, choose to walk along the lakeshore, where you will meet Elyana. If have the animal liking on, you can proceed after a couple meetings with her. If you have an dorm added to your house, you can eventually ask her to move in.
For her saving sister quest after she moves into your dorm. First you need to talk to lazarus about the monster she mentions for the horn. Follow whatever requirements he has. I think it was an outfit request (can't remember totally as it has been a bit). As for the oasis, talk to the soren artist dude there and he will tell you his requirements to tell you the location of the needed monster.
A bikini is needed-You need to go to the bathhouse to the male side, naked, and do fun stuff. You can get the skimpy bikini. I can't remember if you get it right away or if you need to visit the women's bath and talk the cum lover chick. Afterwards you go to the oasis and pose in the bikini for the artist dude to get the location. From there you go home, go to Elyana's room and talk to her. There is a cutscene, you then automatically go there and battle.