The pics help. I guess one question is, what do you want it to look like, or what is your goal? Are you wanting to create some particular sort of habitat, such as for song birds?
I think the natural inclination is to over-clear areas like this, I know mine is. But that isn't "healthy" in wildlife habitat terms. Obviously, the more city-park like you make it, the more you will be inclined to spend time there, but it comes at a price to habitat variety and species support, plus it takes more maintenance, primarily mowing, to keep it in a park-like state.
If it were mine, I would focus on getting the vines under control and aim long-term at trying to create/maintain three habitat zones, ground cover (including rotting logs), bushes and small trees up to 15 feet tall, and maturing, tall trees. A few 1/4 to 1/2 acre clearings maintained by alternating areas bush hogging every year or two in late July are good too.