Some more selected teams.

In Brazil, my team of choice is definately Internacional. Formed in 1909, they actually were one of the first teams to bring in blacks and are also known as "The People's Club". Inter's history is quite full of silverware as well, ranging from the 40's, to the 70's, to the defeat of a great Barcelona and Manchester City teams in international competitions. Recently their successes have continued as Inter won the Libertadores Cup in 2006. But not only that; they went on to defeat Barcelona to become Club World Champions. They also have a very good youth training system.
Now in another South American powerhouse, Argentina,  my favorite team is probably Newell's Old Boys. One of the two great clubs from Rosario, they hold the select honor of once having all of their players represent the Argentine national team at the same time. They've won quite a few national champions as well as being the club that developed Leonel Messi between 1995 and 2000!
Across the world meanwhile, in Japan, my selected team is the Kashima Antlers. Their titles since joining the league in 1991 consist of: J1 League champions in 1996, 1998, 2000, 2001, 2007, 2008, and 2009. They also won the League Cup in 1997, 2000, and 2002. Also, they won the Emperor's Cup in 1997, 2000, and 2007. Finally, the Antlers were A3 Champions Cup winners in 2003.
Meanwhile, not far away, in South Korea, my favorite team is probably Suwon Samsung Bluewings. They were founded in 1995 and even as they were just beginning they were making a name for themselves. They have since became one of the most successful teams both nationally and domestically. At their first try to in the Asian Cup Winners’ Cup tournament they got all the way to the final only then losing. Their good performances have essentially continued from then on.
Finally, into Eastern Europe now to look into Romania. The team I would support would be Universitatea Craiova. Founded in 1921, they are very successful domestically as well as holding the distinction of being the Romanian team to go the farthest in the UEFA Cup, in the 1982-83 season. Despite this history, these days they are a mid-table team for the most part and succumbed to the lower divisions for several years. They've fought back every time with success.