I have heard about people paying lots of money to become Russian cosmonauts, essentially space tourists.
Is it still possible to 'buy your way' into space? What sort of training would be required for a "space tourist"?
I think you are looking for Space Adventures Ltd. They handle the "tourists" on Russian Soyuz flights to the ISS: http://www.spaceadventures.com/ This is the only business currently capable of actually sending you to space for cash.
Space is commonly defined as starting at the Kármán line. I believe you can buy a ticket for an upcoming flight via Virgin Galactic to cross this line on a suborbital flight today. In addition, you can also purchase a much more expensive, orbital trip, from Space Adventures.