There are so many good bars and restaurants in Skiathos town i could not possibly reccomend any one. I think it is just a case of explore yourself and find one to suit!
for restaurants i would say stay off the main strip around the harbour and get among the narrow streets of the old town. There are some fantastic restaurants in there and the prices are quite a bit lower than those on the strip.
For a memorable night, try the windmill restaurant. Set high above the harbour it has fantastic views over Skiathos town and the port. It has a wonderful menu. It also has it's own website to give you a taste of what is on offer!
http://www.thewindmillskiathos.com/The%20Windmill%20Restaurant%20Skiathos%20Greece.html Music bars are in abundance! For nightclubs, try the strip leading away from the bus terminus around the harbour towards the airport.
There are piano bars, jazz bars and much more. I like the Admiral Benbow. It has a bit of a quirky character. just standing outside during the day reading the notices has me in stitches!
As i said, everyones tastes are different. What i like, you may not...simple! Just go and explore and you will find your own "magic" place.
Enjoy Skiathos....................i'm back for my 4th visit in September!

John

lAexandros taverna as mentioned in earlier posts, and most tavernas and hotels/restaurants around the English area. (If you look at Papadiamantis St from the harbour, its up the road to the left of it.