Transportation and stay in Barcelona

My husband and I are in our mid 20s and this is our first trip to Spain. We will be in Barcelona for about 4 days. We will be flying in to BCN airport in August and thinking about staying in an AirBNB in the Gothic Quarter or El Born neigborhood.

1. Is Gothic Quarter or El Born neighborhood a good choice and close to most "must see" things?

2. Is a taxi the best way to travel from airport to Airbnb? About how much is a taxi from BCN airport to Gothic Quarter?

3. What is the best way of transportation while staying in the city from place to place?

4. What are your top reccomended restraunts or bars?