I like to travel, but I wish I could stay with someone at my destinations. Weird? Or feel the same?

not weird at all! there are organizations devoted to this kind of travel, where you can stay at peoples homes, sometimes even for free. One of the better known of these is Servas (check their site http://www.servas.org/indexen.html) - I have a good friend who always uses their services, and in exchange he sometimes hosts people at his home. Check out the wikipedia link for all kinds of this kind of traveling.

So bottom line - there are millions feeling this way and doing it, so definitely NOT weird.

Check out couchsurfing.com, hospitalityclub.org, couchexchange.org (if you're a touring musician), or womenwelcomewomen.org.uk (if you're a woman). They're all sites that hook up travelers like you with people who like having strangers crash on their couches or in their guest bedrooms. Of course, you have to be careful. Meet with the people first, and if you feel at all uncomfortable, find somewhere else to stay.

a bit odd