It's difficult to find any direct materials online with references for America's Got Talent. From their [official website][1]:

> *DOES IT COST ANYTHING TO AUDITION?*  
> Auditioning for America's Got
> Talent is completely free. Sure you still need to get to the venue,
> and maybe get a hotel for the night if it's a long drive, but we
> cannot reimburse you for your expenses. But if anyone is claiming they
> are with the show and want to charge you money to audition please
> contact us right away.
> 
> If you make it beyond the first round of auditions, we will discuss
> with you in detail how we reimburse for travel arrangements as you
> progress in the competition.

So clearly once you actually get *onto* the show, there are some travel arrangements. I'll cheat a little in my answer here, and rather than give examples of America's Got Talent, I'll post some examples of information I found regarding American Idol, which will likely have a very similar expenses policy.

For American Idol, [Fox Business][2] describe the process for auditioning:

> *Auditions Can Be Expensive*
> 
> ...Audition costs remain a financial
> hurdle for hopefuls who follow the Idol audition trail from city to
> city, hoping to be selected.
> 
> "What the TV audience doesn't see is, if you're from San Diego and you
> audition and make it through in St. Louis, then you have to return to
> St. Louis a month or two later for the next round, and then again for
> a third round," says Rushfield. "Some can't afford to and drop out.

They then describe actually being a contestant:

> Once chosen for the top 12, Idol contestants are provided room and
> board, although the accommodations have varied widely from season to
> season.
> 
> "There have been years where they want to show it on the air so they
> put them up in mansions in the Hollywood hills; other years, they've
> put them up in this apartment complex that's not seen on the air,"
> says Rushfield. "It's nothing fancy, but it's not squalor. They have
> roommates all the way through, and when their roommates get cut, they
> consolidate them to save on the rooms."
> 
> A far bigger expense falls to the contestants' families. "They don't
> provide travel or expenses for families," he says. "That's the expense
> that causes the most trouble for Idols and their families."

What about costume expenses? The same article states:

> *Ever wonder whether Adam Lambert paid for those outrageous "Glambert"
> costumes he wore on the eigth season? The answer is yes -- and no.*
> 
> "They go on a shopping trip with a fashion consultant who works on the
> show and they get like $450 a week to spend on whatever they want,"
> says Rushfield. "But a lot of them dip into their own pocket because a
> few hundred dollars doesn't go that far in a lot of these expensive
> stores in Hollywood. The clothes are theirs to keep after the show."

As for how much they earn overall:

> *How much does an "Idol" finalist earn?*
> 
> "Their payday for a year of very hard work comes to something like $1
> million," says Rushfield. "The top four all do Disney World
> commercials; that's one of the biggest paydays. They also get money
> for albums and shooting the Ford commercials."
> 
> Then there's the show itself. The American Federation of Television
> and Radio Artists, or AFTRA, pays each top 24 contestant a performance
> fee: $1,571 plus meals for two-hour Idol shows, $1,303 for one-hour
> shows and $910 for half-hour results shows. But they all must fork
> over $1,600 to join AFTRA.
> 
> The biggest payday for most of the top 12 is the summer tour. "They
> get around $150,000 for four or five months of touring," says
> Rushfield.
> 
> The higher the finish, the larger the paycheck -- with notable
> exceptions. According to Forbes, season four winner Carrie Underwood
> led the pack, earning $13 million from June 2009 to May 2010. She was
> followed by season one winner Kelly Clarkson with $11.7 million. But
> Idol runners-up Chris Daughtry (third at $10.2 million), Kellie
> Pickler (fourth, $7.6 million) and Jennifer Hudson (sixth, $3.5
> million) routinely out-earn winners David Cook, Taylor Hicks, Kris
> Allen and Ruben Studdard.

A different [Fox Business][3] article gives a more granular view of parts of the audition process:

> Only a handful of contestants get called to audition for what the show
> calls "Hollywood Week." An "American Idol" insider confirmed that
> contestants are flown out to Los Angeles and housed at the show's
> expense. No meal money is given, but the shoot days are usually rather
> long and a contestant could easily eat all three meals for free while
> on set, according to the insider.
> 
> While Hollywood Week takes about five days to film, a contestant's
> time commitment varies. "They can stay the whole time if they do
> really well," says the insider. "Or, they can be done within a few
> hours on the first day."
> 
> If the contestant continues to impress the judges, he or she will be
> called back for live shows several months later. Again, the show pays
> for travel and lodging, but the commitment can last several weeks.

To confirm the overall amount earned by an Idol participant in a year, The [American Idol Net.com][4] fansite states:

> *ARE AMERICAN IDOL CONTESTANTS PAID FOR BEING IN THE COMPETITION?*   
> Once
> a contestant reaches the Top 10 and in some seasons the Top 12, they
> start to receive a paycheck for appearing and performing on the weekly
> shows. At that point, they are also required to join the AFTRA union
> (The American Federation of Television and Radio Artists). The most
> recently reported hourly rate I could reliably source is $1,571.00 per
> two-hour episode and a bit less for a one-hour show. The Top 10
> contestants (plus the wildcard pick this season) who go on to perform
> on the American Idol Summer Tour will also receive a fee of $1,000.00
> per show. 

> *ARE THE FINALIST’S FAMILIES COMPENSATED FOR BEING IN THE
> AUDIENCE?*    
> Even though American Idol loves to utilize the finalists’
> friends and family sitting in the audience, either by airing reactions
> or actually bringing them up on stage, they do not compensate them for
> their expenses. The cost to the families can be quite high considering
> they must pay for travel and accommodations out of their own pockets.
> The audition process can also be a financial drain, considering some
> hopefuls must travel long distances to get to the audition city. If
> they make it through the first round they must travel back again to
> audition for the next round. These rounds of auditioning can be
> separated by several weeks. Many believe it’s well worth the expense
> for the chance to become an overnight success. 

> *DO THE CONTESTANT’S
> CHOOSE THEIR OWN WARDROBE AND WHO PAYS FOR IT?*    
> The contestants are
> given a clothing allowance. The last verified amount I could find is
> $450.00 per week but it must be increased occasionally to keep up with
> inflation. A stylist, employed by Idol, accompanies contestants on
> shopping trips to help choose outfits for the coming week and advise
> them on what “look” will work best based on the their individual style
> as well as the style of their upcoming performance. Idol also provides
> them with a hair stylist and makeup artist.

So, I've given a bunch of articles and I'm well aware the majority of them aren't actually focused on your particular question - about America's Got Talent. However, I'm hoping they will still be of us considering the fact the shows are very similar and thus very likely have similar expense regulations.

  [1]: http://www.americasgottalentauditions.com/faq/preparing-for-your-audition/
  [2]: http://www.foxbusiness.com/personal-finance/2012/03/01/cost-being-on-american-idol/
  [3]: http://www.foxbusiness.com/personal-finance/2011/05/06/american-idol-contestants-story-625057601/
  [4]: http://americanidolnet.com/american-idol-faq/