If you 'take a shine to' someone it means you like a person from the moment you first met or that you favour them in some way. When you 'take a shine to someone' you usually feel you have a good rapport and understanding of the other person. The feelings you have for the other person might be romantic, professional or friendly. Similar in meaning to 'take a liking to' someone. This is a casual expression used more often in informal contexts.

1. I didn't like Tom when I first met him, but recently he and I have been getting on quite well. I think I'm starting to take a shine to him.
2. It seems like Julia has taken quite a shine to you! Maybe you should ask her on a date?

1. I immediately took a shine to Sophie. I think that she has a lot of potential which is why I have agreed to mentor her.
2. Alex didn't like the new layout of the shop at first, but since then he's taken quite a shine to it.