Eventually, you'll have to decide what you want out of these miles. Traditional road touring is a great way to knock out the miles. It's pretty easy to put 50-80 miles in an average road touring day. I've done a lot of road touring and for the most part, it's fun enough, but in some places it can be outrageously boring. Imagine stretches of road like those through Kansas that are strait and flat for hundreds of miles. Ick.
There are more and more people getting off the pavement and folding in more off roads. From where you are, you could do a nice loop of a few days using a mix of paved and gravel forest roads. For that, you could use a touring bike or mountain bike. If you want to fold in any singletrack, that will be a mountain bike.
Regarding trailers, that's just one way to haul your kit.