I've gathered metal ore from a mine, but the forge only uses iron bars to create items. I can't seem to find an option to convert my ore into bars; do I need a special kind of forge, or I am missing something obvious?
What you are looking for is a furnace/smelter, which is typically in the shape of a very large, cone-shaped oven. Not all forges have a smelter, but Whiterun has one.
A list of all smelters can be found on this site