All the orbital launch-sites I know of are to some degree also military bases, like Cape Canaveral and Plesetsk, and they have in any case restricted access. However, which of them can I visit, without breaking any law? I am most interested in a manned launch, if that helps to narrow down the question. (although I am then left with only Russia and China.)
The United States launch sites are the most accessible, both in terms of location (near population centers and accessible) and political environment. Kennedy Space Center in particular is welcoming of visitors, and to a lesser extent Wallops, and you can get into Vandenburg occasionally if you ask nicely.
As far as other countries, the major centers are Baikonour, which doesn't allow tourists (Yet), Guiana Space Center, which does have some launch site visitation (But is in a fairly remote part of the world anyways), and the various Chinese sites, all of which seem to allow some form of public visitation.
Bottom line, the US and China seem to be the most open about visiting their launch sites.
I'm not sure if it'd be exciting enough for you, but I visited Tanegashima with a friend in January 2013. It's a bit of a way off the west end of Japan, but not quite as far as Okinawa. They mostly do satellite launches and tracking. (this qualifies it as an orbital launch site for you, yes?)
Once you've made the ferry, hydrofoil or plane trip to get there from Kagoshima, it's easy to get around. Tanegashima Space Centre is super welcoming and they run guided tours of the facilities as well as leaving you free to walk around the sizeable public areas. TSC is a beautiful place, and they're justifiably proud of it.
JAXA's website has info in English on visiting and booking tours. Wikitravel has good details on travelling there. Note that the guided tour around the facilities is all in Japanese, but don't let that stop you. For what it's worth, you'll have the opportunity to get eyes on the launchpad from about... 1km away? Maybe a bit further; I don't recall the bus route so I can't confirm that on google maps.
It is certainly possible to visit Kennedy Space Center and see a launch. This link gives viewing sites for KSC, Vandenberg, and Wallops as well as a link to information on purchasing tickets for on-site viewing.
Unfortunately, since you wish to see a human spaceflight launch, KSC is not going to fit the bill for quite a while.