According to this leaked road map that Engadget got a copy of it looks like AT&T will be the first out of the gate with the new Samsung Galaxy S4 on April 26th with T-Mobile hot on their heals on May 1st and Verizon May 30th.

While there were rumblings of the AT&T launch date before posted by AT&T themselves sadly all links that I found are now dead. One thing that is known for sure is that by the time you read this you will be able to pre-order the device. According to this sheet it can be pre-ordered today April 15th! Sadly there are no mentions of price on any of these documents but it is expected that AT&T's prices will be as follow 16GB $199 and 32GB $249 respectively, with a two-year contract.