First trip to Spain.

You planned a trip before--I guess those were cancelled, but are you starting over totally from scratch?
When you say shoulder season, do you mean fall or spring?
If ten days, you could split it between Barcelona and one other town. That's what I would do, but I am less of a city person.
I would start with a guide to Catalonia--that would be my first stop. happy travels!

Barcelona has plenty to occupy all of your 7-10 day vacation. If you wanted to spend 2 nights elsewhere, the historic cities of Girona or Tarragona would work.
The beach resort of Sitges would be a break from the intensity of Barcelona.

These 3 cities can each be reached by train from Barcelona in 40-75 minutes.

Check out the human towers at a meet in Barcelona, called Castellers --

https://www.barcelona.de/en/barcelona-castellers-human-towers.html

Thank you all for your suggestions.

Barcelona is a huge transportation hub with cheap flights all over Spain. For me 7 to 10 nights is way too long for Barcelona. I would do 3 or 4 on the end of the trip as that sounds like where you're flying out of. I would think about flying to Seville or Granada for part of that time. Much depends on age and interests. J

I am taking my first trip to Barcelona in October.

The Hop On Hop Off Bus gets good reviews . I plan on doing it.

I am hoping that I will get to Montserrat. Sre you familiar with Montserrat?

You may know that The American Cup is in Barcelona from now but especially from August through October. Lots of activities around The American Cup.

I guess my answer depends on how you like to travel. We went to Barcelona for 2 weeks back in May. We found enough to do in Barcelona to enjoy ourselves the whole time we were there. Some days we didn't do anything in particular, just wandered around and ate tapas. But we did take a day trip to Montserrat. It's an easy train ride, and you can spend a whole day there. If you're spending a week or 10 days in Barcelona, I strongly recommend Montserrat.

I don't think the HO/HO is a good idea in Barcelona for people without mobility issues. The public--transporatatioj system is good and taxis are not expensive. I saw the HO/HO stop at Placa de Catalunya when I arrived in Barcelona in 2015. There was a very long line of people waiting. They were not all going to fit on one bus, and there was no bus in sight. The HO/HO could waste a lot of time if you hoped off very often.

In 2018 we spent 6 days in Barcelona and 3 in Sitges during shoulder season.
Day trip to Montserrat. All day event.
Loved Sitges. 28 minute train ride on the R2sud. We watched a castellers contest- great fun.
Plenty to do in Barcelona for 6 days.
Don’t waste time traveling.