Hey Travel-b
Such a great city, so little time hahah! You will find yourself on a busy schedule I´m afraid but don´t worry, it is totally worth it!
To start with the journey from the airport to the hotel (you did not give an address but I am assuming you are talking about The Hilton on Av. Diagonal 589-591): The train leaves the airport every 30 minutes or so and you should get off at Sants Estacio -also called Barcelona Sants- from here you can directly take the green metro line up to ´Maria Cristina´ which is closest to your hotel (about 5-10 min walking max).
If you do not feel like taking the train and metro you can also take a taxi from the airport which will cost you between 20 and 30 euro depending on the traffic.
http://www.barcelona-tourist-guide.com/en/airport/barcelona-airport-transport.html
I know you want to avoid tourist holes but I can tell you, some are worth it! Check out Parc Guell and Casa Mila -also called La Pedrera- two of my favourites.
You could take one morning to walk from Plaza Catalunya towards the beach so you can have your lunch there. I would advise you to avoid the Ramblas when you do this. You might walk on it the first bit just to see what it is like but then, as soon as you can go the right or left into the smaller streets and find your way down to the beach through these. They are usually less crowded and you have a good change of walking into a small local market.
Halfway the Ramblas in these sidestreets you can find the Boqueria, which you should definitely visit. Budget tip: buy some fresh food here and eat it on the beach so you do not have to pay for restaurant food.
If you go to see the Sagrada Familia be sure to look for Avinguda Gaudi which lies directly next to it. Follow this street till the end and you will end up at the stunning Hospital St. Pau, be sure to go inside, it is free and very beautiful!
As for going out: the real nightlife (as in clubs) starts ridiculously late here, not before 2 at least. There are many smaller bars with good music that are open earlier but for the real clubbing you will have take a whole night, so maybe not the best idea for a three day visit. I do not know any jazz/blues bars but I am sure there are plenty to be found. Just search google 
As for food: if you want to save on money then avoid the restaurants in the direct neighbourhood of tourist attractions. usually when you walk two streets away from them you will be able to find food that is at least as good but for a better price. Most traditional tapas bar - restaurants do not look really appealing from the outside but the food is usually good!
I went to a restaurant in the born district called La Luna, great ambiance, delicious food, reasonable price:
http://www.lalunabcn.com
If you want any more information just let me know!
OHannemarie
(Oh-Barcelona)