Tallima: My wife's biggest dream is to backpack Europe for a month or so in summer. We love to see the things nobody else sees. No Sistine chapel, but the local coffee shop would be fun. And no tour guides, but real people. And maybe see the Sistine chapel too.
We wouldn't be able to go for a few years, and when we do, we'll never be able to afford it. So my wife's dream is crushed. But I told her all about you people and said, "I bet we could stay at goglodytes houses and see the real Europe!"
So if we were to go in a few years, can we stay at your house?
"real Europe" => I didn't know there was such a thing as "fake Europe" :o)
Joke aside, "postcards" are real, it's just that pretty pictures are taken on sunny days or that we put on some traditional clothes only 1-2 times per year but it's still part of the "real" life. Just for teasing if you have a twitter account:
https://twitter.com/MagicalEurope Anyway, what you're looking for is "couchsurfing", there is a dedicated platform to it
https://www.couchsurfing.com/
I did some couchsurfing around 2000-2003 (that term didn't exist back then, it was just people meeting through forums). So from experience, you'll have to accept that you see things "partially" because of location and time constraints. Also, as I guess that you won't be coming just for the week-end, youtr host will probably be unemployed or a student otherwise they won't have time to drive you around during the week on daytime.
A cheap alternative are the youth hostels in every major cities but they're highly sollicited and well as it's collective dorm, it's kind of an acquired taste (I personnaly don't like them).
Also, you imply you'll travel through countries. If you wanna see the landscapes during your travels, take the train (it's also much less bothersome than planes which requires various check-ups). You can get a good discount on the ticket by getting Interrail passes as they're specifically tailored for people travelling through Europe:
http://www.interrail.eu/
Don't take the international bus lines, they're cheap but they highly suck.
Most importantly:
Your perception of Europe could highly varies depending of your background. Like if you go to Germany, there will be a huge perception difference if you're coming from Texas or Maine as landscape and climate are closer to one state than the other.
You should decide in advance what you REALLY wanna see of Europe because it highly varies from one region to another and depending of your likings. To take the example of France: you can go to Paris and do the usual tourism stuff, going in bars, etc. BUT you'll get into totally different worlds and lifestyles if you vistit the various countrysides like Alsace, Brittany or Corsica as each ones have their own customs, activities, landscapes, history and you have more chances to get to know the locals than in cities.
Attached pictures are kind of what you'll remember from your trips (well, that's what I remember ^o^) so if you have time/money constraint, you'll have to make choices instead of "trying to do everything at once" and ending up dissapointed because you didn't take the time to enjoy what your surroundings could offer.