I'm all for this KONY 2012 thing and the reason behind it, but please don't take the video everyone is sharing just as it is. As good as it is to spread the message of international child trafficking (sexual, warfare, labor), Kony isn't a threat anymore. He and the LRA haven't been an issue in Uganda for over half a decade. In fact, the worst person in Uganda is their own president, Yoweri Museveni.
If you are interested in Uganda specific issues, injustices that are actually currently relevant are things like their anti-homosexuality bill which, until recently, included the death penalty. Also, late last year they granted amnesty to a former LRA commander, which doesn't mean the LRA will be starting again in Uganda, but it is a terrible thing for the victims of the LRA looking for justice. Also, the Ugandan government has recently been restricting the voice of the people by banning public protests, threatening, intimidating and, without just reason, arresting journalists. Unfortunately because these issues are somewhat new to this area, there aren't any charities or organizations I'm aware of that target these issues specifically in this region, but there are many many many international LGBT and Freedom of Press/Freedom of Speech organizations.
If you are interested in the issue of child trafficking, this occurs (sadly) everywhere. There are a number of charities that target this specific problem. Some of the better rated ones in terms of how well they handle their money and how transparent they are (two things that I've heard invisible children has some issues with, though I can't say I know much about it) are:
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International Justice Mission-
Free the Slaves-
American Jewish World Service-
Girls Education and Mentoring Services Just as a heads up, the American Jewish World Service does not deal singularly with human trafficking, but I thought they deserved a shout out since they have really great business practices. Also, I do not know anything of GEMS business practices, but I do know, however, that when I was in Amnesty International people would always tell me "What about the people in the U.S.? You don't want to help them?" and so, that's a specifically U.S. organization and they even have a pretty alright documentary out about their work on netflix instant watch, I believe.