So you are going to start the bananas inside? If possible, why not start them outside in a pot for now? It's early August. As the first frost approaches, bring the plant gradually into the shade and then inside. Bananas will survive a whole winter in total darkness with no water whatsoever. I did this with my cavendish bananas last winter. Anyways, back to your post......

I was wondering how one would go about starting small banana's inside under mostly artifical light?
In the winter, you can do this.

There is a small window that allows some natural light, but I am sure that I would have to have some additional light source.
Not neccesarily, the idea for the winter with low light is to keep the bananas alive. Granted, if you want them to grow, and it will be minimally you will need some light and or sunlight.

Any tips for inside growing? What kind of light would work, would Florescent work or would it have to be more robust then that? What about watering?
Any light will do, or you can do away with light totally if you wanted to, but I will not go there, since you want to have your bananas inside to view and admire. Sunlight is best. Find the sunniest window possible and place banana there. Be sure to rotate banana every week so that the leaves get an even amount of sunlight. Regular light sources such as a lamp will do. I personally would not invest a bunch of money in grow lights, since your banana will be outside next summer. The key is to keep the banana alive. Growing inside over winter isn't going to happen on a large scale. As for watering bananas inside during winter, this part is KEY.... Water very minimally. Too much water will cause the banana to rot. A good rule of thumb that I found with me in the winter is to stick your finger in the soil. If the soil is dry up to your second knuckle on your finger then water. Also, do not fertilize in the winter. When summer comes, and the bananas are outside and growing they love tons of water and a good fertilizing. Hope this helps!