Eixample is a convenient first-time base with broad streets, Modernista architecture and excellent metro links, though the district is large and your exact location matters.
Ciutat Vella, including the Gothic Quarter, puts historic Barcelona within easy walking distance but can feel crowded and noisy. Gràcia has a more local, relaxed character and suits longer stays, while still connecting easily to the centre by metro.
Barceloneta works well when beach access matters most, but it is busier in summer and less central for some major sights. Tip: Compare the nearest metro stop, evening noise and air conditioning as carefully as the neighbourhood name.