Will my fillings change color, too?

If you have fillings or caps, they don't change color.  Only natural teeth whiten.  There are anecdotal reports of extreme strength bleaching protocols changing the color of fillings, but these regimens also cause extreme sensitivity, requiring significant prophylactic doses of pain medication. 

A more sensible approach is to whiten the teeth to the desired shade, then replace any fillings that no longer match.  Replacing the fillings before whitening is not recommended, as your final shade is not entirely predictable. 

If I'm getting veneers anyway, do I still need to bleach?

In order for veneers to look their best, they should be translucent, (show light though them) just like natural teeth.  That means they tend to show through the color underneath.  You should always whiten all the surrounding teeth anyway, so that you can have a brighter smile overall.