Barcelona is highly walkable. I used the Metro a few times to reach more distant locations during a long stay in Barcelona, but I mostly walked. Locals have commented here that taxis are reasonable. I don't know whether Uber operates in Barcelona.
Your tickets to Parc Guell will include a shuttle bus from one of the Metro stations, so it will be easily accessible. You must buy those tickets in advance because none are sold at the park. I don't think tickets sell out very early, but entries are timed, and if you want to go at a particular time (such as first thing in the morning), it would be prudent to buy earlier. I don't know how early would be "safe" for that purpose.
La Sagrada Familia tickets are also timed, and the earliest time slot is popular, because it means you'll walk into a church that isn't already full of folks who had earlier tickets. Again, I don't know how early the most popular times sell out. I used the audio guide and thought it was good.
Other places for which you'd want to buy tickets in advance--if they are of interest--include: Casa Mila/La Pedrera, Casa Batllo, Picasso Museum (feel free to skip it; it's often unbelievably mobbed) and the Palau de la Musica Catalana (because the English tours may sell out). All of those places except the museum are modernista buildings.
Another modernista place you might like because its architecture is lovely: Sant Pau Modernista Site. It's a multi-building complex that takes at least 2 hours to see. You should be able to walk right up, buy your tickets and go inside. It's not terribly far from La Sagrada Familia.
Otherwise, I recommend walking around the Eixample area, where you'll encounter some other modernista buildings whose exteriors you will appreciate.
I also liked the walking tour of the Barri Gotic conducted by the tourist office. It was very informative, and they limit the number of tickets sold, so you won't be part of a huge contingent of visitors.
Girona is a good side-trip, but I wonder whether you'll have time for it with only three full, non-jetlagged days in Barcelona. I doubt that you'll want to leave. There's a lot to see in Girona, so I wouldn't recommend a "quick tour". Realistically, I'd figure on a full day by the time you get to Sants train station, ride to Girona, get from the Girona station to the medieval area, see the two historic churches, walk around the wall, wander the streets, have lunch, and get back to Barcelona.