As US American (which I assume you are) you can not stay in the Schengen zone (most of continental Europe) for more than 90 days in each 180 days, you can also stay in the UK/Ireland, also 90 days in 180 or even 180 days in a year, and there are other countries where you can also stay outside the Schengen zone.
But you can not work in any of those countries.
If you are from Canada, Australia, New Zealand or one of a small group other countries you might get the option of getting a working holiday visa.
That will allow you to work in one country and staying there for up to a year. Not that it is easy to find jobs so you should not count on earning while traveling.
Traveling like I describe above costs in the range of $US 100 per day, the actual amount depends on where you actually stay and how the exchange rates evolve.
Other options for places to stay include camping (but carrying a tent and all needed gear around is not much fun,) rented apartments or rooms in private houses and often airbnb is named for that, of couch surfing where you stay for free in the homes of people and you promise to host in the future often you do cook or bring a gift for the host.
Those places to stay are almost never central in the cities you visit, taking extra time, which is not as much a point when you travel long term.
You need to bring all you need but not too much.
It is easiest to travel with no more than a day pack, say about 10 kilo, 22 pounds, or less, but first time travelers mostly take 3 times as much in a big backpack and a day pack. Often the day pack is a small backpack that is worn on the front, very heavy with books. Not comfortable.
While a gap year sounds great, for many people it is the end of education, as after a year of holiday it is very hard to get back in the routine of studying. And tasting travel often makes that you want to do more travel.
And get a copy of Lonely Planet's "Europe on a Shoestring" which is aimed at backpackers and budget travellers. In addition to the usual tourist information on each country, cities and towns, and tourist sights, the book includes information on how to eat cheaply, lists hostels and cheap hotels, health and safety tips, and lots of other useful information.
You also need to be over 18 to stay in any accommodation.
In addition to the very good detailed answer from Willeke, this site provides a lot of useful information on backpacking in Europe: http://www.backpackeurope.com/
Backpacking -- tramping, trekking or backcountry camping -- is essentially the combination of hiking and camping in the backcountry. A backpacker carries camping gear: a tent, sleeping bag, cookware, food and clothing, in a backpack and hikes to a backcountry camping destination.
Backpacking trips range from short one-night trips to multi-day trips. Some trips start at one trailhead and end at another. And some backpackers even set-out on months long distance end-to end treks called thru hikes. Popular thru hikes include the Pacific Crest Trail (PCT) and the Appalachian Trail (AT).
But to get started backpacking you don't have to walk thousands of miles. There are many short and moderate destinations that are scenic and beautiful.
Now that you are interested in going backpacking let's get prepared for your adventure.
Getting in Shape for Backpacking
If you are a first time backpacker, or heading out for the first time in the season, make sure to get in shape before you hit the trail. Backpacking is more difficult than hiking because you are carrying the added weight of your camping gear.
To get in shape for backpacking, start hiking with low mileage and carry a lightweight pack. Build up your mileage and add weight to your backpack as your trip gets closer. The more fit you are for your backpacking trip, the better you will feel when you are on the trail.
No time to train? It is understandable if your backpacking trip is just around the corner and you haven't done much training, but make sure to lighten your load. Take only essential and lightweight gear, and consider choosing a destination that is only a few miles from the trailhead.
So you are in shape for your trip, but what should you pack in your backpack?
Backpacking Gear
The goal of most backpackers is to keep their pack light, but still carry all the camping gear they need to make their trip comfortable.
Ultimately, you only need food and shelter for a successful backpacking trip. There are a few essential backpacking items that each backpacker will want to carry and a few items that a group of backpackers can split up to share the weight.
Before you get packed to go, check our backpacking checklist to make sure you haven't forgotten anything and try to leave non-essentials at home. Each pound you shed from your pack will make your hike easier and more comfortable.
You are packed and ready, now where should you go?
Where to Go Backpacking
National and state parks, wilderness and forest areas are popular backpacking destinations. Check with the ranger station in your region for popular routes. And your local camping and outdoor retailer should be a good resource for books and maps.
Look for a destination near a creek, river or lake so that you have a source of water. Once you have selected a destination make sure you obtain the appropriate permits and check the regulations for food storage, camping and fires.
Now that you selected a destination, what precautions can you take to stay safe in the wilderness?
Backpacking Safety
Do you have a map and compass or a GPS device? And do you know how to use them?
Always let someone know when you'll be gone, your destination and route. And make sure to call them when you return.
A small first-aide kit is an essential item to bring along on any backpacking trip. Also know what your emergency resources are in the region you'll be backpacking. In a wilderness emergency, remain calm, determine an action plan and seek help.
Now you are all set to go on your backpacking adventure, but do you know how to keep the wilderness wild?
Backpacking Ethics
The Leave No Trace Foundation is a non-profit organization that has a set of values and recommended ethics for campers and wilderness travelers. Most backpackers agree that you should "leave no trace" and "pack out what you pack in." The Leave No Trace core principles include:
Dispose of Waster Properly
Leave What You Find
Minimize Campfire Impacts
Respect Wildlife
Travel and Camp on Durable Surfaces
Be Considerate of Other Visitors
Also, make sure to check with the park or forest service ranger station for regulations specific to the area where you will be camping. Depending on the region and time of year, special regulations may not allow campfires, may require specific food storage containers, and sometimes specific areas are closed for restoration. It is generally recommended to camp at least 100-feet from water. Following regulations, and core backpacking ethics helps to conserve the wilderness for generations to come.
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