A car is of no use in any major European city. And you need to stay in the city to facilitate efficient sightseeing.
The tourist situation in Barcelona has changed massively in recent years. Visitor levels have risen so sharply that there are at least six sights for which you need to get tickets in advance (if you want to see them, I mean): Picasso Museum, La Sagrada Familia, Parc Guell, Casa Batllo, Casa Mila/La Pedrera, and Palau de la Musica Catalana. I was last in Barcelona in 2016; for all I know there may be additions to that list.
The need to pre-purchase tickets, which in most cases must be timed tickets, makes it necessary to do more planning in Barcelona than many of us like. On a long trip, you can plan to do one of those popular sights first thing each morning, then you can be flexible with the rest of your day, spending as long as you want at each place. On a shorter trip--and both of you are planning on 4 days (which I'm betting is really 4 nights, so only 3 days), you are going to be putting together a jigsaw puzzle if you want to go inside the top sights rather than just observe their exteriors.
How much time will you want at Sight A? How long will you spend in line at Sight A even if you already have a ticket in your hand? How long will it take you to walk or otherwise get to Sight B? Will you want a snack or lunch on the way? How long will that take? You need to answer all of those questions to know what time you'll be ready to enter Sight B so you can buy that ticket in advance. Yes, it's going to be that difficult if you're interested in a bunch of those six sights.
I urge you to read your guide book and really think about which places you truly want to see. It will be easier to plan your visit if you go to only some of the six I listed and add on other places than can be visited spontaneously. I cannot tell you what to see, because people's interests are so different. That's what guide books are for.
I do recommend that only people really interested in Picasso go to the Picasso Museum. The place can be as crowded as the Vatican Museums (Google for pictures), and I find it hard to enjoy an art museum if I can't get close enough to the art to read the labels. Barcelona has many other very good art museums that do not generally require pre-planning. I'm partial to the Museu Nacional d'Art de Catalunya, which has some very interesting exhibitions of modernist decorative arts and medieval frescoes from churches in the Pyrenees as well as the expected paintings, and the nearby Miro is great for modern-art lovers. But there are many more. It's not the Picasso Museum or nothing.
If you scan back through the Spain forum you'll find many earlier threads about sightseeing in Barcelona. They will answer questions you haven't even thought to ask, and they will give you the perspective of a bunch of different travelers.
