Some of you guys are missing the point of the OP - he WANTS to learn a local language. I'm sure he/she knows English is widely spoken and understood and isn't asking because he's worried about surviving. I
t's great that you're interested, and in my experience it's always fun and very much appreciated by locals when you can speak a local language or dialect. It's great in Uganda because you can speak English if you have to, so there's no pressure to communicate, but using some Swahili is great fun and the locals will be extremely happy and appreciative.
I'd pick swahili, particularly if you want to arrive already knowing something. There are loads of tribal and local languages all over Uganda, and it's very difficult to get proper training in those ones unless you're on the ground there, or if you can find someone in your community where you live now (which is possible, but tricky). There's way more info on Swahili to be found at home so you can arrive already with some knowledge.
I've used the "Teach Yourself" series before for Hindi and Farsi, and they're quite decent. I think they have a Swahili course too. Get some of that under your belt then put an add on Craigslist for a Swahili tutor. You can probably find someone from East Africa (I don't know where you live now, mind you) and get him/her to teach you expressions that are local so you'll actually be saying things the way people in the area do (that's important).
When I was in Uganda I had about 25 phrases and questions in Swahili: "How are you?", "Where do you live?" "What's your name?" that I used on the kids, and I could also ask for places to find food and drinks, hotels, etc. I never ever found myself in a place or situation where no one spoke English, so speaking Swahili was mainly for fun. But it was well worth it, and people got a huge kick out of it and were always happy to teach me new words and phrases (another good one to learn is, "What's this called in Swahili?" as you point to this and that). It's rare that travelers bother to learn Swahili or any other local language, so you'll make a good impression on people and be a good ambassador.
Have fun! You're on the right track in my opinion.