Biking the country was my first (and only, so far...) attempt to travel, and it was definitely the most rewarding thing I've ever done. I didn't make it very far due to somewhat poor planning and my buddy abandoning me after only a few days, but in 14 days I traveled from Cornwall, NY (an hour above the city) to the top right-hand of the state, then west to Scranton, PA. Unfortunately, I got stuck there and caught a ride back home shortly after. I'm planning on taking the whole next warm season to do it again, and trying to get to the west coast.
I'm definitely not an expert from a measly 14 days, but here's a few things I learned/was told by bikers I met:
1. pack lightly. make some panniers out of plastic cat litter tubs and secure them to your bike rack so they don't bounce when you ride. if you're doing this in the spring/summer, you really only need the outfit you're wearing, some extra socks, an overshirt, a tiny hiker's tent and sleeping bag, and a couple big water bottles (I think poland spring makes some particularly huge ones. try to refill them in deli/grocery store/convenience store bathrooms). you really don't need much more than that, and it makes it a whole lot easier to get up hills and rough trails. i ended up mailing almost everything i had home after 3 days. it's not necessary, but a camel pack is pretty great to have, too.
2. get familiar with basic bike maintenance (ie learn how to change a flat - it's hard as hell if you haven't practiced- and pack some tools to tighten things up.
3. get in great shape BEFORE you leave. my friend figured he'd scrimp on training and build up endurance as we went, but he ended up getting sick and hurt after only 5 days and headed back home.
4. start looking for somewhere to sleep 2 hours before the sun goes down. sometimes it's really difficult to find somewhere safe to stash your bike and set up
5. don't pack a lot of food. just some grain and dried food or even dry dog food. finding food as you go along isn't too hard
It's definitely liberating, and I felt like I could go pretty much anywhere at a reasonable pace without having to compromise much. I haven't hitchhiked or hopped any trains yet, so I can't really compare the two, but it's definitely a great experience on it's own.