General searches, for example in Google Maps (search term "billiard parlor") will find lots of businesses that have pool tables but are completely lacking in any real interest in the game. There are some chains like "Dave and Buster's" that cater to the very casual player (unless they've changed since the last time I was in one) and are not worth visiting. You could just go from Google maps and stop at every room along your way and ask for local recommendations if you don't find quite what you are looking for. If banjo music starts, move on.
Regional pool magazines are one way to find interesting rooms. Here is a site that has magazines for three regions in the US. Each magazine has weekly/monthly schedules of tournaments. In addition, there are ads from the more active rooms that have tournaments and leagues.
http://www.onthebreaknews.com/
There is a tournament calendar on the AZB front page
http://www.azbilliards.com/
There are five or six good rooms in the New York City area, including Steinway and Carom Cafe.
Stieinway, NYC
Carom Cafe, NYC
Amsterdam Billiards, NYC
Sandcastle Billiards, Edison, New Jersey
Fargo Billiards -- Fargo, North Dakota -- perhaps the nicest room in America but a little out of the way
Hard Times -- Bellflower, California and Sacramento, California
Jointed Cue -- Sacramento, California
California Billiards -- Fremont, California
Family Billiards -- San Francisco, California
Greenleaf's -- Richmond, Virginia
Diamond Billiards -- Midlothian (near Richmond), Virginia
Freezer's Ice House -- Tempe, Arizona (near Phoenix)
Kolby's Corner -- also in Tempe
Griff's -- Las Vegas, Nevada
Chris's -- Chicago, Illinois
Q-Master -- Virginia Beach, Virginia
There is also the site
http://www.findsnooker.com that attempts to list all of the snooker tables in the US. Usually if a pool hall has a snooker table it caters to the more serious player.