I don't really even know what anime genres typically consist of so I'm not sure. As you can see I liked Chobits so I could always start there. In general I would probably enjoy something cute/fun (so no action or fighting series), not overly sentimental and with a good pace so I don't get bored.
Sorry I can't be more helpful, there's just so much in anime that it all kinda goes over my head. Honestly any suggestions you think are good would be very appreciated, cause then I can 'dip my toes' so to speak, and go from there
If you're looking for good romantic/comedy I'm going to suggest K-ON!, CLANNAD, and Oriemo.
Azumanga Daioh and Kuragehime are also really good but Azumanga makes a lot of jokes that a person relatively new to anime/Japanese culture probably won't get and Kuragehime is about otaku/NEETs that again a newer viewer may not understand. Not that they can't enjoy them, rather they would just be more enjoyable to watch if you had more information.