Hi Juli,
As @acraven mentions, that's located in l'Eixample, a classy Modernist area with wide streets and plenty of beautiful late 19th-century buildings. The area, as much of the rest of the city, is totally safe (yet noting you're in a big city of course!).
To go to the airport, your best bet -other than taking a taxi (aprox 35€, depending on traffic)- is to walk to the train (not metro!) station in Passeig de Gràcia, located across the street from Casa Batlló, merely 10' away on foot from your apartment (http://bit.ly/2flV8Tf). If you're catching the train (35' to the airport), you can use the T-10 travelcard and then the journey will cost you only 1€ each. The T-10 travelcard for 1 zone is a card with 10 journeys that allows you to move about in the city on different means of transportation: bus, metro, tramway... and it's rather cost-effective: T-10 = 9.95€ vs single ticket=2.15€. More on that: http://bit.ly/bcn-t10 IMPORTANT: the train will end in Terminal 2 at the airport, if you're going to Terminal 1, there's a free shuttle bus (green) from there that connects both terminals. If that's your case, plan for an extra 20' or so... as you'll have first to find the stop, wait for the shuttle and then the ride to the other terminal (http://www.aena.es/en/barcelona-airport/connection-between-terminals.html)
The day you go to Montserrat, you'll need to go to Plaça Espanya (plaça=square in Catalan language, pronounced "pla-sah"), located in the west side of the city. For that, from your location simply follow the same instructions than above to Passeig de Gràcia, but instead of the train, take the metro line L3 (green) to Plaça Espanya station (~40'). Once there, you need to get out into the square and look for the entrance to the FGC train (also subterranean) which is to your right, at 100 yards or so... the entrance looks like this: http://www.trenscat.com/metrobaix/images/P080630006.jpg Easy peasy! There are other ways to travel to Plaça Espanya from your location, and you'd probably scrap a few minutes off the journey, but not knowing the city you're better off not having to transfer or anything, that's why the metro is probably the easiest way for this.
SIDENOTES
There are different means of transportation to the airport: Aerobus, bus, train, taxi and most recently also metro. You can always buy single tickets for it but if using travel cards there is a bit of complexity for this particular journey:
- T-10 travelcard: you can use that if going to the airport by train, either from Passeig de Gràcia or from Sants stations but NOT by metro (line L9). It also covers using bus #46 to the airport, which departs from Plaça Espanya. It does not cover Aerobus.
- HolaBCN! cards: you can use this card on the airport-city journey on all means except Aerobus.
Obviously none of the above can be used in taxis :)))
Note however that the metro L9 line which connects the city to the airport serves the northern part of the city so it's unlikely to be useful to most visitors, which tend to stay elsewhere. Other parts (ie southern and central areas) are already well connected to the airport by other means.