The one going straight to the basilica is Gaudí Avenue, named after Antoni Gaudí, the architect who designed the Sagrada Família (as well as other landmarks like Park Güell, Casa Milà / La Pedrera, or Casa Batlló); it was designed to connect the two landmarks of the Sagrada Família and the former Hospital de la Santa Creu i Sant Pau (today a UNESCO world heritage site).

The one in the background is Diagonal Avenue (no, really), one of the main thoroughfares in the city, intended by Ildefons Cerdà (designer of the Eixample) to cut through his grid layout together with Meridiana Avenue (which roughly follows the Paris meridian, or rather the Barcelona-Dunkerke one; there's also the perpendicular Paral·lel Avenue, of course, though sadly they don't cross), crossing at the Plaça de les Glòries Catalanes, which Cerdà intended to become the new city centre (alas, the Plaça Catalunya, some 17 blocks to the south, ended up taking that role).

From Cities Skylines experience it's usually to relieve traffic blocks by providing a direct path to areas/landmarks that have a higher than average traffic load. Not sure why they did it though.

I can't tell if they are actual streets, pedestrian-only areas, or bus loading/unloading zones. There look to be structures along them that could be market booths or buses.

That's if you mean the two blocks with the diagonals going through them. If you mean the one in the back that's slightly off-angle from the grid, my guess for that is that the road existed before the modern city did and wasn't removed to create the grid. Or it might be a rail line.