My brother and I spent 2.5 great days in Barcelona, enjoyed a lot of the incredible sites, and did it all relatively inexpensively. Here is what we did:
Day One
Stroll La Rambla, admiring the sites, shops, and hoards of people.
Stop along the way for some incredible fruit cups or (overly sweet) 1 Euro smoothies in La Boqueria.
This is a great place to try a few different snacks, but it’s quite touristy so may be over-priced. Take time to admire the stacks of chocolates, dried fruits, cheeses, meats, empanadas, and more. 
We didn’t eat there, but my mom highly recommended Bar Ramblero, to the left of the entrance in the market for some incredible fish and potatoes.
Continue your walk down La Rambla until you find yourself at Plaza de Cataluna. Cross the plaza and take the right hand street straight ahead, which is Paseo de la Gracia. Follow this until you pass the Casa Batllo (Gaudi building). 
The architecture is absolutely incredible, but it´s a little bit pricey to go inside (20 Euros/ person). We admired it from the outside.
Continue up until you pass Casa Mila (another Gaudi creation) on your right. This one is less impressive (IMHO), but you can head into the lobby and admire some of the work from the inside. Again, we did not pay to enter.
Shortly after you see the last house, you will be taking a right and finding your way towards La Sagrada Familia, which is another 20-25 minutes from the Gaudi house.
We hadn’t purchased tickets to enter, since we only wanted to see it from the outside, but if you plan to enter, its a very good idea to purchase tickets ahead of time, which gives you a specific entry time. Tickets start at 15 Euros/ person and the price increases as you “add” things, like visiting extra parts of the cathedral or having a guide.
After seeing La Sagrada Familia, head towards the Barcelona Cathedral in the Gothic Quarter (in Pla de la Seu), another architecturally impressive and massive cathedral.
Since you’re already there, meander in and out of the winding streets of the Gothic Quarter, stopping along the way for drinks and tapas.
Day Two
Our only “full” day in Barcelona, we used this day to head to Park Guell. But first, stroll along the port and admire the glistening water, swooping birds, and people out and about basking in the sunshine.
Stroll along the sea and then hook back to Plaza de Colon and head up La Rambla.
Make another pit stop in La Boqueria for another fruit cup. They were incredible!! Also…admire more of the yummy food…
Properly fueled, you can start the trek to Park Guell.
*The walk from La Rambla to Park Guell is very, very long (over 1.5 hours), but if you´re able, it´s a great way to see more of the city and enjoy the sites. If not, there is a hop-on, hop-off bus that heads there, as well as a way to get there via metro.
(this house above reminds me so much of Cuba)
Follow basically the same route as the day before: La Rambla – Paseo de la Gracia – Gaudi houses – head in direction of La Sagrada Familia – keep going alllllllllllllll the way up to Park Guell. 
Park Guell is absolutely huge, with most of the “park” being free.
If you want one of the iconic photos with *the* mosaic wall, you will need to pay to enter.
This is the wall from afar.
The ticket isn’t too pricey (8 Euros/ person or 7 if you buy it in advance). We chose not to buy the ticket, and instead hiked around the beautiful park, admiring the breathtaking views and nature.
Gaudi really knew what he was doing with the beautiful use of mosaic and tile. It made me want to redecorate something with those beautiful tiles.
Eventually, we headed back down the hill, stopping along the way to look in the mierda shops with lots of souvenirs that model the Gaudi tiles. I wanted a mug, but never found the perfect one.
We had an end goal of heading back towards La Rambla, but made our way slowly since we stopped for a café con leche, then shared a cup of deliciously creamy gelato.
Then, since we weren’t quite sugared-out, we stopped for churros con chocolate so Lee could try the deliciousness that is thick, molten hot chocolate that you dip greasy, chewy churros into. 
They weren’t the best I’ve had, but they did the trick.
Once you have your merienda, head back down Paseo de la Gracia to La Rambla and into the Gothic Quarter to see the spooky cobble-stone streets at night, all lit up from the shops and restaurants and bustling with energy.
Stop for tapas and drinks in Plaza San Josep Oriel II at the place called PaTapas where you can get some amazing pimientos de padrón, tortilla española, and more- all for great prices.
Day Three
We had an early afternoon train, so kept our ambitions low. A great activity for a half day is a climb up Montjuic mountain near the beach that offers incredible views and a really cool cable car ride down (if you pay for it).
The walk up to the initial “base” only takes about 15 minutes, and another 15 minutes of steeper climbing brings you to a castle and some seriously awesome views.
After that, head back down, grab yourself a coffee and a bocadillo, and prepare for a relaxing journey home 😉