What's the most cost effective way to get animation?
It really depends on what you mean by "cost effective".
South Park uses a small team of animators who design entire 30 minute episodes every single week on computers using basic pin animation, whereas The Simpsons animates an episode over the course of a few months, using a huge crew of animators, assistants, and in-betweeners from South Korea. South Park is comparatively "cost effective".
The truth is, original animation - which I'm actually starting to study more seriously now - is really hard to do.
But if you're not too worried about quality, and you're not worried about having some other company's watermarks all over your video, I'd say that by far the most cost effective options for amateurs would be to use sites like www.goanimate.com or www.xtanormal.com.
These sites allow you to animate basic scenes with pre-built characters using text-to-voice software, your scripts, and a very basic user interface. Animations like this can be produced in a few hours.
Another cost effective - and more original - way to make simple animated videos is to record screen-capture videos of virtual sims, like Second Life, The Sims, or World of Warcraft; or to record video games like Halo (a la "Red vs. Blue") or Call of Duty using your own voice over combined with the characters from the games to create your scenes.
Of course... These options are really just cheats.
If you want to create actual animation with original characters, and you don't have the skills yourself, the best way to keep the costs down will be to find a young animator and keep all the designs as simple as possible, and limit character motion/interaction to basic things like talking and moving arms. Complex backgrounds, environments, moving shots, and really detailed characters - and a lot of different characters - take an amazing amount of time and skill to produce, and are quite expensive for that reason.
"You get what you pay for" is exceptionally true in this medium.
Note: I'm currently reading a great book by Richard Williams (who animated "Who Framed Roger Rabbit?") called "The Animator's Survival Kit". I highly recommend it if you want to learn more of the real process of making drawn characters come alive.