Do I need a Schengen Visa (as a US citizen) to backpack throughout Europe?

[Adam Z]

I'm a US citizen, planning on going backpacking throughout Europe this coming summer (probably sometime in early June to late July or early August), and I'm unsure what the exact requirements for a Schengen visa are? Do I need to apply for one? Or am I going for a short enough amount of time that it is not necessary?

And if I do need one, does anyone have any recommendations for which country to apply for it in (as in, which is cheapest), since I'll be traveling without the continent anyway?

Thanks!