RC Miguel Angel is a good hostel to stay near one of the 3 big metro stations (Sol)
In Madrid, depending on how far ahead of time you intend to book, it will cost you anywhere from AU$30/per night and upwards (more like $40 for decent hostels)
One thing you should visit there is the El Tigre Tapas bar. It is 2 blocks North-East of the Gran Via metro station (2 north and 1 east).
Great place where you can eat until you are really full for extremely little. It is packed with locals and tourists who you can easily find people to go out on the town with. over 10 years ago
I've stayed in Los Amigos backpackers hostel a number of times and would definitely recommend it. It's cheap (dorms ~17 euros) and central (right between metros sol and opera). It has good facilities and is a great place for meeting people.
http://www.losamigoshostel.com/ about 10 years ago