It once took us over an hour to get from the gate to Madision Junction. It can be like rush hour on a bad day. I think if you got in really early you would beat the rush.

I assume from your post you aren't interested in camping--that's my favorite option actually. As such, I don't have hotel or lodging recommendations because of that, but if you are interested in staying in the park, I would go for the Canyon area. It's centrally located, and therefore easy to make excursions to all parts of the park.

For camping, if interested sometime, I like the campgrounds at Canyon and Grant's Village. Both have showers, and are reservable. And they're very nice.