For a very nice experience closer to the old section of Barcelona, check with the Hotel Neri. It's lovely and in a very nicely redone (with a modern twist) historic building.
For a very nice hotel in the Eixample neighborhood (which is a rather modern neighborhood and within reasonable walking distance (if you are big walkers) of La Sagrada Familia and several of the popular Gaudi buildings, then check with The Alma Barcelona.
Or, you can do as we did, and stay two nights at each.
Since you mentioned loving food, check out the Food Lovers Tapas Tour
http://foodlovertour.com/tours/
We totally loved the tapas tour, which stops at three different places: two local tapas restaurants (that we would have never found on our own), and then the third local restaurant stop is an entire meal. The guide explains the food and drink (all included) that you experience at each stop. Warning: You will NOT go away hungry. In fact you will have that after-Thanksgiving feeling. While it is not inexpensive, we really got a great value, discovering the interesting places filled with locals, trying some very interesting and good food, learning about the food and dining traditions, and all adult beverages (or other choices, if someone does not drink) included, plus a fun guide to help you find your way around and back. Seems we had about 8-9 people on our tour, including the guide. I don't eat seafood (my husband loves it), so I just let them know when I booked and while there was plenty of seafood for those who loved it, there were also more than enough choices for me, too.
On our group, three were around age 40, but one maybe late 20s, and three around age 50-60. You can see photos posted on Trip Advisor that show some of the foods and some of the photos of the typical people who go on these tours. It really was a highlight of our trip to Spain, and it's a fun, casual evening with a group of interesting people who love food and love meeting others:
https://www.tripadvisor.com/Attraction_Review-g187497-d5953405-Reviews-Food_Lover_Tour-Barcelona_Catalonia.html
Have fun in Barcelona!! Three is a lot to see/do in just four days (which was exactly the amount of time we were there...we could have enjoyed more days).