Hi, milleramanda53! I am an illustrator who would love to get into illustrating children's books. I am like you in that I am not schooled in digital illustrating software. I think you could very easily avoid the "techno-trap" by simply creating your illustrations with traditional media and then scanning them to create digital images of your work. I do this all the time, and the results are wonderful!
As far as what art medium to use, I think there are several that would meet with your approval. I am a watercolorist, so I would first recommend it. Many people are daunted by the prospect of using watercolor, but I think their fears are largely unfounded. Of course, the style of your work would have to fit, but good line drawings done in waterproof ink could easily be painted in vibrant washes of watercolor. Artist's acrylic would also give you the vibrancy you are seeking, as well as would pastels.
I hope this helps. I'd love to see your work sometime. If you ever need an illustrator for your children's books, please be in touch.