Around the stream are some beautiful trees, but unfortunately they're
being chocked by vines.

Chopping these at the base of the trees is fine. They can grow back
if it turns out you really wanted them. Before I went too far I'd
identify your vines. English Ivy is probably the most common here
(Washington, DC), as well as porcelainberry, grape, poison ivy,
virginia creeper, and some others. Some of these will really take
over, but others are much more benign and maybe just need to be
trimmed back a bit if they are in the way.

Also there is just a tremendous amount of underbrush in the area that
makes the entire area unwalkable and unusable.

Again, identifying the plants might help. Few people will suggest you
would desire to keep something like multiflora rose (because it is
non-native, thorny, not particularly attractive, etc). But amongst
that underbrush might be some plants which aren't really taking over,
and would be quite attractive if not covered in vines (or whatever).

I second the comments of everyone else about treading somewhat
carefully and being careful not to end up destroying the streambank
and ending up with a bunch of mud sliding into the river. But
starting by trimming everything back isn't going to hurt anything.
You can always let them grow back as you figure out which ones are
good and which ones not so good, where you want to put a path, etc.