What are the most popular areas to do private pilot flight training in the United States?
The FAA has a document on their website which breaks knowledge tests down by FSDO rather than state, and it gives a good idea of where the most training is taking place.
It is called: 2013 Airmen Knowledge Tests by Region and FSDO, and lists the following Regions/FSDO's as processing the most knowledge tests (in descending order)1:

Southern 28,333
Orlando 9,996
South Florida 5,958
Alabama &amp; NW Florida 4,680
Atlanta 3,882
Tampa 2,504
Western Pacific 18,416
Scottsdale 6,053
Riverside 1,556
San Diego 1,463
Sacramento 1,460
Long Beach 1,327
Fresno 1,270
Los Angeles 1,142
San Jose 1,121
Southwest 15,943
Dallas 2,731
Fort Worth 2,581
Houston 2,438
Oklahoma City 2,347
San Antonio 2,321
Eastern 14,552
Richmond 1,956
Greensboro 1,366
Farmingdale 1,280
Philadelphia 1,198
New York City 1,069
Boston 1,041
Great Lakes 11,860
Fargo 2,120
Dupage 1,414
Indianapolis 1,358
Minneapolis 1,142
East Michigan 1,111
Northwest Mountain 9,784
Denver 2,205
Portland 2,131
Salt Lake City 1,964
Seattle 1,611
Central 9,019
Nashville 4,293
Wichita 1,304
Alaskan 812

Based on this data, I would say that the areas surrounding the Orlando, Scottsdale, South Florida, Alabama &amp; NW Florida, Nashville, and Atlanta FSDO's are the most popular training areas, with the Southern region being the most popular by far.

1 To improve readability, all FSDO's with less than 1,000 tests have been removed.  See the source data for the complete list.