Step 1: Join PostCrossings
This is free, but you will need an email account (not unlike this fine site).
As a personal note, I suggest you mention your hobbies and the kinds of cards you would like to receive on your profile. This helps people sending you a card to select a card that best fits you interests. I also suggest a face picture, but understand that some people feel uncomfortable with that.
Step 2: Buy or make some postcards
You can be creative and follow some of the instructables on this fine site to create your own post cards. Be sure to stay to the standard size (4" x 6" in the US) to improve the chances of your card being delivered successfully around the world.
On the other hand, you can go to the "drug store" (Walgreens in my area) and buy some post cards featuring local monuments or locations (like the post card of the Gateway Arch shown here). You can also choose funny or sexy postcards (as one of the people I sent a card requested) since the cards in your country are likely to be quite different from cards in other countries.