You seem pretty experienced at traveling already, but here's what I'd recommend:
Avoid carrying stuff you can't afford to lose.
Barcelona (so I've heard) is a wonderful place, I am envious. I spent 3 months traveling through Europe, but as bad luck would have it, thanks to an airline delaying my luggage, I had to skip visiting Barcelona. But also Barcelona is the #1 pickpocket capital of Europe. This shouldn't scare you though, it's much better than #1 mugging capital, but you're gonna have to stay alert, especially in crowded places.
There's a funny Colombian saying that goes "No des papaya!" which literally is to "Don't offer/give out papaya (fruit)" and it translates to "don't make it easy for a thief, i.e carelessly flash expensive things, or do things that mark yourself as a potential target for the thief."
If you must carry a day backpack, make sure it's small, or at least don't overpack it. For a day of just walking around the city, I prefer small "gym sacks" that have straps, and just hold stuff like an extra cell battery, cables, map, water, sun screen, etc. Leave your passport at your hotel/rental. In crowded areas carry your bag in front of you. And be wary of strangers that get too close.
Always carry a paper copy of your passport, and scan and upload important travel documents to the cloud. Additionally, you can even create shortened url that points to a cloud-shared item in case you need a copy of something in an emergency (i.e like for example, all my contacts are in my cell/wallet, but if my cell/wallet gets stolen, I'd go to the nearest internet cafe and browse to like https://goo.gl/ + WdcAid and get the info I need).
Avoid carrying so much stuff. It weighs you down, you get tired quicker, and you risk attracting more attention to yourself. Since you say you're doing a short-term rental, and no one is supposed to go in to clean the apartment while you're there, you can leave electronics you don't need there.
In a hostel or hotel or temporary housing, be creative and hide your stuff in plain sight
I think you have a pretty safe place for your things. If you want to be extra safe (or paranoid like me), hide your money or important things in unexpected places, or hide them in plain sight.
For example, I've always used an empty deodorant stick container, and roll up most of my cash I travel with ($500-600 max) inside it and place it among my other toiletries in the bathroom (if not shared of course). No burglar is going to look through there. You can use a ziplock bag and put your passport or other important documents in the refrigerator for example. Just don't forget where you're hiding your stuff!
Safety in and out
Please be safe when abroad. Don't drink too much, or do anything you wouldn't normally do at home. Be aware of your surroundings and overtly friendly strangers at bars, etc. I like to also implement some of the hotel security tips from these guys to make sure my room is safe. It might be overkill, but it never hurts. When out and about, make sure you've planned your route or know more or less how to get there, that way taxi drivers don't rip you off, and you don't look like a helpless tourist holding a map upside down. The site WikiVoyage.org is an amazing resource of info on just about everything you need to know. And they have an offline mobile app, you can preload using wifi.