Hello!
A couple of random thoughts:
You do not mention WHEN exactly you plan to visit and if 5-6 days are days on the ground or include arrival and departure days. Also on a small time frame, one extra day makes a difference, sop is it 5 or 6 days?
If you are experienced on traveling solo you may know that meeting people to hung out with often is pot luck. Good food can mean many different things but most taverns in Greece tend to serve items based on fresh ingredients and costs on most low key places are very reasonable. Sea food tends to be expensive. Any place with decent pernament population and/or lots of visitors offer a decent nightlife but this varies according to season and destinations. Mellow bars where you may enjoy a few drinks and conversations till early morning are often more common than clubs on most places.
The easiest islands to connect from Athens are obviously the Argosaronic ones, Aegina, Hydra, Poros, Angristri... Spetses could be added to on this list.Those are 2+ hours ferry ride at most from Piraeus. Often they sort of operate as suburbs of Athens due to close proximety and might be very busy on weekends. Each one has a different mentality though, for example Angistri often attracts people who love free camping or Hydra more upclass visitors or Spetses the yaught club crowd...
Your dates and budget an time of arrival might be crucial to advice on a few more ideas. For example arriving on Athens airport you could get a flight straight away to an island, no island is more than 50 minutes flight away... Or you may time arrival to the airport with a transfer to Rafina port and a quick ferry ride to an island no more than 4-5 hours ferry ride away...
I think an island such as Paros or Naxos might fit the bill for a first timer. Those suit the generic idea most people have for a Greek island (sandy beaches, white/blue architecture, harbour towns, nice taverns and bars, boat trips in season, decent bus network to explore around some of the key spots and so on)... Larg-ish pernament population, so lively even on shoulder season and not feeling like a tourism theme park either, while depending on specifics, prices are quite reasonable.
You may get a flight or a ferry to Piraeus and save your Athens exploration for the end of the trip.
That said, Greece is not only about the islands. Nafplio, a very pleasant sea side town with lovely waterfront and a medival castle on top, is only 2 hours away from Athens and is a great base to explore the surrounding area too.
Galaxidi, a town with long shipping tradition, interesting captains' mansions and a lovely small fishing harbour is about 3 hours bus ride from Athens and you may combine it with Delphi UNESCO Heritage site.
Hiring a car for a couple of days, there are tons and tons of mainland destinations to explore, providing often a more real picture of Greece than many islands provide...