It's all pretty easy. Every country seems to have their Craigslist equivalent where you can find apartments to stay in for a short term (a month or two). I've spent a lot of time in hostels in the past, which are a fun and cheap way to live as well. If you're new to long-term traveling those would be a good option to get you started because you meet lots of interesting people. But after a while you kind of need your own space and living in a 20-bed dorm stops cutting it, regardless of how cool the people there are.
Since I rent rooms in apartments they usually have their own kitchens. I cook for myself or go out with friends to eat.
The rent varies greatly from place to place. I've been told by many that Munich is the most expensive city in Germany. I rented a big room in a rather small apartment in a great part of town (next to Ostbahnhof) for 150 euro a week with one roommate That seems to be about the right price based on the other offers I found. In Belgium, I lived in a small town just south of Brussels. I rented a room from a retired couple who lived in a 3-story country house with a huge, beautiful garden and massive kitchen for 100 euro per week. They were frequently gone so I had the house to myself. In London there's a website called Gumtree that's like Craigslist. I found a room for 160 pounds per week. It was much harder finding a room in London than anywhere else because everything is outrageously expensive and everyone seems to want a minimum 6 month stay (I'm only here for 3 weeks).
In Belgium you can get a bottle of 12% Trappist for ~2 euro; in Germany a liter of beer for ~6 euro and a pint for ~3 euro; in England a pint for ~3 pounds. Food at the grocery seemed cheapest in Germany, where I never found myself spending more than 15 euro in one visit that would last me a few days of eating well.
So, after you convert euros and pounds to dollars, you can see that it varies a lot. In England in particular the conversion rate is killer--something like 1.6 dollars to a pound right now.
My advice would be if you really want to do some long term travel, just buy your ticket and do it. It's really extremely easy to figure things out once you arrive somewhere. You can email me at my username at ucla.edu if you want any more info.