Regarding concept art in the games industry; before becoming a lecturer one of my roles was creating environment concept art for the game developer I worked for. To me I would recommend you aim for either a Concept Art degree or a 3D Games Art degree; Illustration work (having also worked as an illustrator) tends to favour style over technical skills - not that it can't be technical, but it's usually more about creating an eye catching image than demonstrating knowledge of anatomy or how perspective is constructed. Some concept art can be for that purpose as well; but the majority of it is things like multiple designs of the same prop, vehicle, monster, character, environment, or is things like paint overs of the 3D level that the designers have put together and then they ask you to redraw their screen grabs to make them look more interesting - so adding more foliage, different lighting ideas, random props etc - whatever it takes to make the scene feel more alive or scary or charming (depends on the game!).
If you pick a Concept Art course make sure it teaches a lot of 3D modelling as well - that is a vital skill for a Concept Artist. You will need to show the same character / creature / environment from different angles, and so making a reasonable base mesh in ZBrush or Maya or Blender is vital - you then paint over that in Photoshop.
As a concept artist you need to really understand anatomy (both human and animal) and how objects are constructed - vehicles, guns, props, buildings and also work in perspective. You need to be interested in replicating surfaces (metals, rough stonework, skin, cloth, plastics, glass etc) so you should show this in your work.
You will need to show style frames / splash images as well - those are the ones where you can go to town and show off your compositional and lighting skills.
If you want to see what my first year students made last year, one of my colleagues made a nice video showing their work (this is art from across all four degrees, including Comics and Concept Art students and 3D Games Art students)
https://vimeo.com/1027745113 This might give you a better idea of the variety of work you might make in future.
Good luck, keep drawing and check your proportions and work from references!