There is actually a surprisingly wide range of different types of entertainment in Second Life.
Given that places come and go, no landmarks will be provided here. Instead, this notecard will cover a few of the types of entertainment that exist. To find places, make use of search: open it with Ctrl-F, then type in search terms and see what you find. Look not only at places, but also event listings and groups.
Clubs are very likely the most popular type of entertainment in SL. There are a huge number, scattered all over, and offer a very wide range of music and dancing for all types of audiences. Go to a venue, find the dance system, start dancing, and just socialize.
There are live performances all over SL, be they signers and/or musicians, and in many different musical genres.
Quite a few places offer more formal dancing for those who enjoy it. Some require fully formal attire, some are a bit more relaxed. The more upscale places often require membership as well. A few have attractions other than just dancing, such as fine dining.
Yes, there are restaurants in SL. Naturally you don't consume real food, but you can simulate the experience. Restaurants range from diners to upscale luxury establishments that require formal wear. Some also have space for dancing, and may feature live singers and/or musicians.
Art in many forms thrives in SL, again, in a very wide range of styles. One popular form is dubbed "Not Possible In Real Life" (NPIRL), in that it makes use of the fact the objects do not naturally fall, or can be made to glow or emit particles. The Linden Endowment for the Arts regions are a good place to go, with regularly changing exhibits.
There do exist live performances of plays within Second Life. These tend to only run for a short period.
Yes, believe it or not, there is live ballet and opera in SL. Given how much is involved in setting up such productions, they are often one-off events.