HDMI will give you the best picture quality from your Xbox. Depending on what TV you have, it will either be 720p or 1080p, anyways your xbox will pump out what it can pump out, and your TV will display what it is capable of. Can always go into your xbox settings and make sure that it is putting...
If you want to game tablet is out of the question. If you are looking to have mobility for school and going home the laptop could be viable solution. There are many gaming laptops out there with specs similar to your current PC.