I've recently gotten a bunny, less than a week, and I researched a lot before getting it. I bought a large cage, timothy hay, Zupreme rabbit food... As soon as I brought him from the shelter, I filled his water bottle, his food bowl, and the hay disposal. On the first day he was really scared, so he spent most of the day on his hiding spot. But on the second day, I noticed he ate quite a bit of hay and drank the water. However, I feel like he isn't eating much of the pellet. As soon as I put cabbage or carrots, he eats them straight away, but I feel like he is barely touching his food. How much are they supposed to eat? Am I over reacting? He is 5 months old and 3.7 pounds
There is a very good chance that the pellets your rabbit was being fed prior to your adopting it was different. As long as he is eating hay and drinking water you are fine. I would try to make sure he has as much hay as he will eat. See this answer on another question for more information on how to best transition your rabbit to the new feed.
If you stop giving him treat foods eventually he will start eating pellets more regularly. Right now the pellets are not something his gut is used to so it may take some time for his belly to adjust. Keep free feeding him hay, and make sure hay and pellets are the only foods he has to eat. Once he starts eating the pellets regularly I would start treating with Dandelion greens and/or parsley. Those are nice easy foods for the rabbit and ones he will love.
Your bunny is from your local shelter (great choice!) and 5 months old, he is just becoming an adult. As a healthy adult rabbit pellets are not required in his diet as long as Timothy hay is always available and sufficient leafy greens are given daily.
If feeding pellets a 1/4 cup per 5 pounds of body weight is recommended, so your bunny would be getting about a two table spoons a day. This guideline is a starting point, you will want to monitor your bunnies weight and if they are getting skinny or fat adjust the pellets accordingly. See this answer for pictures and details of an extreme case.
In our house we feed pellets twice a day, half in the morning and half in the evening. Hopefully your rabbit as access to run in the house a couple times a day or more if you let your bunny run in the morning and/or evening, and feed them their pellets when it is time for them to go back to their room (if caged during the day), simply put the pellets in a bowl, call them, and they come running, the pellets and the bunny go into the cage. Even if you leave the cage door open this is great training, as you will always be able to get your bunny to go in his cage when needed.
You mention feeding carrots and cabbage, neither of this are recommended as significant part of a rabbits diet. There is some controversy around feeding any cabbage to rabbits but, but a little in moderation (like a leaf a day or less) is probably ok. Carrots are high in sugar, any sugary food should be treated like candy, given as treats in moderation.