I placed the raster image, I used Live Trace...
LiveTrace provides a plethora of tracing options. What presets or custom options did you use for the trace?
Did you Expand the LiveTrace results? Did you Ungroup the LiveTrace results?
...and I tried to fill it with color.
Exactly how did you try to fill it with color? Below, you describe the selection tools you are using, but not the clicks, commands, or tools you are trying to use to fill the selection.
First I used the selection tool and ALL paths in the image were selected.
Then I used the direct selection tool and parts of paths were selected.
Using the DirectSelection tool (the white pointer), select one of the paths you want to fill. Now look at the Appearance Palette and see what is selected. Does it say Path? Compound Path?
Open the Document Info Palette. From its flyout menu, turn on Objects and Selection Only. The palette info will then tell you if you have an open or a closed path selected.
I shift clicked other paths to select what would make a closed object
and selected a fill color and that's when I got the fill color in different
places.
Depending upon the tracing options you used, you may have any combination of open and/or closed paths selected when you do that. You need to understand what you are actually selecting.
You also need to understand that selecting several open paths that visually create a closed area, and then trying to apply a fill to the selection does not result in one closed and filled path. It simply applies a fill to the several paths you have selected. In Illustrator, paths can have fills, even if they are open.
The LivePaint feature can allow you to treat merely visually enclosed areas as if they were, in fact, closed paths, and then fill them with the LivePaint bucket tool. But doing that, you would not first direct-select the paths. Again, all this is why you need to describe exactly and completely what you are doing.
JET