It's a good idea to bring new pets straight to the vet in general. It may have been seen by a veryvet at the shelter already, but it's helpful for your vet at the least to have a record of what's healthy for your cat, and also it's possible for the cat to pick things up at the shelter, as it has many cats in close proximity. So take it to the vet and tell them you're concerned the cat has not been eating.
However, there are also cats that are very picky, or like to slowly "graze" on their food throughout the day. I suspect your cat is a "grazer" since "grazers" will often be interested when you first put new food down, and then nibble at it and wander off. In this case, keep trying foods to see if they seem to be particularly interested in one. Also, leave a constant supply out so your cat can nibble throughout the day.
Another trick is to get the cat's attention, pick the food bowl up, turn around so the cat cannot see the bowl, and then set it back down. Cats do not have object permanence, that is, the ability to understand that items out of view are still there, so when you take the bowl out of view, and then set it back down, the cat will think it's a new bowl, and it may then try nibbling at it some more.