My wife and I rode in one on our trip last week. You just hang around the entrance, just past the train station, for one to come over and let off the passengers it has. Easy as pie.
The horsedrawn streetcars, along with all other attractions in the Disney theme parks (excluding arcades) are included in regular park admission. All Main St. Vehicles are one-way trips between the Central Plaza (in front of the castle) and Town Square. There are signs on Town Square noting which vehicle loads at each location. The streetcars load right in front of Main Street Station, in the center of Town Square.
There is no cost to ride any of the vehicles on Main Street. Also there are just two locations where you can get on one of the streetcars. One is by the train station and the other is in front of the castle. There are posts that note when to wait.
Although we've been to MK probably 10 times, my wife and I had never ridden the horse-drawn carriage until our last trip. Right after entering the park after it opened we noticed that the front row in the carriage (next to the "driver") was empty. We hopped in and rode to the castle.
What a fun way to start the day!
Everyone waves at you. Half way down Main Street the Dapper Dans hopped on the back and serenaded us for the rest of the trip.
It was a lot of fun. Not sure it would have been as magical otherwise, but making that our first thing to do on our first day in the MK really started things off nicely.
First, I thought only bumbles bounces. Second, I know that they tend to run in the mornings, especially during the summer months when it gets hot and crowded.