This article is all you need to get started collecting autographs from athletes playing any of the four major sports. Collecting autographs for current players is easiest, seeking signatures of retired players requires a bit more work but is also quite simple.
For each autograph request (often referred to as TTM: through the mail) you will need 2 stamps, 2
envelopes, one letter to the athlete and a sports card or 3x5 note card for them to sign.

Things You Will Need

Stamps & Envelopes: Use self-stick envelopes. It is more likely for the athlete to return the card if they don't have to lick the gluey part of an envelope to return it. Many people would rather not lick untold numbers of envelopes and since you're asking for a favor it is best to make it as easy as possible to get a return. I also recommend purchasing a complete roll of stamps. I send requests regularly and the fewer trips I have to make to the Post Office the better. You will need to include an empty envelope with your address and stamp already on it (referred to as a SASE: self-addressed stamped envelope).
The Letter: I always write my letter by hand, it's much more personal and lets the athlete know you're not just typing a form letter and changing the names. I typically begin my letters with the same opening sentence: "My name is Phil and I am a fan of yours." I make sure to ask a question since some athletes will actually reply with a letter and some answers. Here is an example letter: "My name is Phil and I'm a fan of yours. I think you're having a great season so far and it was great that you got two hits the other day against the Brewers. Can't wait to see you play again! I recently began collecting autographs of my favorite players and it would be great if you could sign the card I mailed. Thanks so much."
I kept the letter short and was sure to include something personal about the player. Statistics can easily be found on the websites of any of the major sports.
The Card: It is most common that I mail a sports card of the player. If I do not have a card for the player and still want the autograph I will simply include a 3x5 note card. The most exciting thing for an athlete to sign is a ticket stub from an event where they played. Every time I attend a sporting event, I mail the ticket stub off to a player who had an exceptional game or to someone who made an exceptional play. I make it a rule to only send one item to be signed.
Current Players VS Retired Players: For current players (and during the playing season), you only need to send your letter to the facility listed on the team's website. These addresses are always available online and cost nothing to obtain. Retired players sometimes still work for teams in some capacity or broadcast in which case the autograph request can be mailed to the team or the radio/television station. For information about retired player's current occupations I typically do a quick Wikipedia search.
If a player is simply retired and is not involved in the sport any longer you will need to send the request to the player's home. This may seem invasive but there are ways this can be done while respecting the privacy of the individual.
I have been purchasing address listings from Harvey Meiselman for a number of years. He sells books for $20-$40 that are filled with addresses of current and retired athletes. Harvey tests these addresses himself and if there are any people who request that they not receive this type of mail then he does not put their addresses in the book. He also emails those who have purchased the most recent copies of the books (released yearly) whenever he discovers that individuals no longer wish to receive this type of mail. As an autograph hound, it is always important to respect each player's privacy and wishes to be left alone. Some players also charge a small fee for a signature and this information can be found in the books Harvey sells.
Successes: To date I have collected 300+ autographs through the mail. My success rate is approximately 40%, so I do 'lose' money to the stamps that go unused as a result of requests that are not returned. This 'loss' is nominal especially since I now ask my friends and family for stamps as birthday or holiday gifts.
I have had players write back to me with not just a signature but a letter answering any questions I asked and thanking me for my interest.
Some former players have written back and sent business cards or other information about businesses they (or family members) now operate.
I once sent a request to a basketball player I was a fan of. In the mail I received a letter from him along with 3 autographed cards! (I only sent him one card to sign). Apparently, he was trying to collect every card issued of himself. He did not have the one I mailed so he kept it and sent 3 others signed as a 'thank you'. The letter explained the trade-off.
Chipper Jones
I mailed Chipper Jones a card and request and he (or someone who works for him) mailed back a signed 5x7 photo. 

This is one of the most personal and fulfilling hobbies I have ever had and I strongly suggest you give it a shot. There is little needed to begin collecting and the pay-off is great when you get those letters back in the mail. After sending off a few requests, getting the mail has never been so exciting.

Have fun collecting!