That shouldn't be too bad at all...350 miles per day isn't a bad number at all.
A agree with moe...my cross country trip had very few destinations. I'd just look at a map in the morning, point my bike in that general direction and hit the road. If I wanted to stop, I stopped. If I wanted to keep going, I kept going.
I averaged 450 miles per day. But I did have a rest day and 3 or 4 days less than 200. My longest day of that particular trip was Steamboat Springs, CO to Joliet, IL, which was right around 1200 miles...around 18-20 hours. Followed by an 800 mile day, then 500 the next so I could have a rest day at an old flame's house out in MA. And boy was it nice to have that day or rest too!
I'm looking to take 2 or 3 weeks off this spring or early summer for a 10k trip down to the southwest.
