Corel has some layout tools that probably put it in between of Illustrator and InDesign, so, without seeing what you mean by "design" it is hard to know.
If it is a Vector Image, an illustration, you can probably just save it as AI inside Corel. Version X7 can export to AI natively to CS6. This way you will preserve layers, that is an important way of keeping things organized and editable.
You can also export images as raster TIF, PNG, JPG, PSD or any format you need.
But again, it depends on what is in the file, and what you need to modify if you need to modify it.
Corel is not a clunky at all. It is a pretty solid application. If it is unknown to you that is a separated issue.
Depending on what you need to modify you could probably learn a couple of tools to do your job. A lot of times it is a lot better to leave a file in its native format. Not all "effects" can be exported if you need to adjust them. Normally they need to be "fixed" for example rasterizing them.
If your "design" is, in reality, a layout template... Just do that from scratch in InDesign. You can probably just export a raster image from Corel and use it as a base to construct your grid.
The reason of this is that in InDesign you need to construct a solid foundation with your master pages and styles.
Corel Draw has its own styling system but you can not export it to my knowledge.