Saving money while backpacking can be achieved in a variety of ways. Of course, the best way is to be as cheap as possible. Your main costs are going to be food and accomodation, so that's what you need to cut down on. Here are some quick tips.
Meet people and network, Work the system, Get discount cards, Get your food at markets
It's very easy to meet people when you're travelling, especially if you travel solo. I've done most of my travel on my own and was never really alone for any length of time. You'll meet people on the trains and in hostels, at museums and so on. Talk to them, share your experiecnes and stories, have a beer, and next thing you know if you're ever in Australia, or France, or Germany, you've got a place to stay. You'll quickly be exchanging email address's and phone numbers with friends you make along the way. Follow up on them, and if you're ever in their neck of the woods you can count on their help.
I mean that in the best possible way. If you can get away with riding the metro for free, do it. If you can get a bus ride for free, do it. If a supermarket is giving out samples of food, take advantage of it. I remember in Barcelona there was one market where they were giving away free kiwi samples in the produce section. Everyday I was in the city, I'd stop there in the morning and get a kiwi for breakfast.
Here are a few good ones that are sure to save you money:
ISIC - International Student Card. Also very good for discounts.
Hostelling International - You'll definitely need this card. Many hostels require it for you to stay there, or they may charge you extra.
Self-catering and buying your own food is a great way to cut costs. Pick up a loaf of bread and some cheese, maybe some Nutella and you're good to go. Drink water as much as possible! Remember though, in Europe the bubbly water is very popular. If you don't like it, make sure you know how to order normal tap water at restaurants or ask for normal water at a store. Many cities have fountains available where you can fill up your water bottle as well.