I've read a little into it and it seems to be so, that some of the new ones are public, while others are 'special' and can only be used if you apply for it (which apparently is very costly).
For example, the .photo TLD is apparently 25.000$ a year and can only be acquired if you have a photo related business. (kind of the same way an .xxx TLD can only be aquired if you're in the sexy-sexy business. .photos, .photography are both public, so I don't see a problem there.
I've decided on getting a .photography one. Pretty dope.
Anyway, Fatcow (my old host; have since deleted my account and got a refund) didn't support them. They're part of EIG (a company that owns MANY big hosting sites, like Bluehost and Hostgator) so I don't know if any other of EIG's hosts will support the new TLD's.
I've registered my new domain at Namecheap and while they do support the new TLD's, I'd rather not have my registrar and host be the same company.