This one is a PS3 exclusive, and was initially offered as a prize for a Hardee's/Carl's Jr. contest. When the first wave of downloadable content is released this costume should be available as DLC, but once again, only for the PS3.
As of launch, this suit is available through per-ordering the game and entering a DLC code. Whether it be the Rhino DLC or some other code, once you enter it you will also unlock this suit back at your apartment. Even though it is labeled as the classic, it is most similar to the suit used in the Sam Raimi Spider-Man movies.
Scarlet Spider Suit
The Scarlet Spider symbol is found beside a bridge in Central Park. If you go near the middle of the park you will see a small car bridge and beside it a red spider symbol. Take a photo of it to gain access to this suit back in your apartment. Note that you can only access this by changing your time setting as mentioned on page one. Also you will need to have progressed beyond chapter 3 of the main game to unlocked the camera.
Big Time Suit
A picture of a white spider symbol in Times Square is needed to unlock this one. Travel there and then head behind the red bleachers surrounded by glass. A quick photo later and you will have unlocked it back in your apartment for wearing around town. Note that you can only access this by changing your time setting as mentioned on page one. Also you will need to have progressed beyond chapter 3 in the main game to unlock the camera.