Discover Card and American Express work on different networks than Visa and MasterCard, and are essentially their own processors. Most credit card processors will assist you with getting set up to accept these cards, but you can sign up with American Express online, directly. With American Express, you receive an American Express Service Establishment (SE) number, which then has to be programmed into your terminal or POS equipment.
Discover Card requires you to work through your merchant acquirer. With the assistance of your acquirer, you will be assigned a 15-digit Discover Network Merchant number, which also needs to be programmed into your terminal or POS equipment. They charge a onetime setup fee of $25.
Both American Express and Discover Card send you statements and bill you independently from your merchant acquirer. This means you will be receiving three separate bills. One from your merchant acquirer, one from Discover, and one from American Express.
American Express do not publish their interchange rates publicly, but once you sign up with a merchant acquirer, (or Discover or American Express if they are the only cards you plan to accept) you will be able to access information about rates. When you sign up with American Express, you also gain access to data on spending trends of American Express holders.