First, doublecheck that the burgundy color grass isn't some sort of ornamental. Take a sample in to a garden center and ask. For everything that's actually supposed to be lawn, the best thing would be to kill what's there and start over with new sod or seeding. That's a big project, but would ultimately yield the best result because it would give you a chance to till up and smooth out compacted soil. A short cut is to overseed what's there without killing the exisiting lawn, such as it is. Chew it up with a verticutter, spread the seed, then go over it again lightly with the verticutter. The result may not be quite as nice as starting from scratch, but it will probably suffice, and it's MUCH less work.