Eagle does have a free version which can be found here,
http://www.cadsoftusa.com/download-eagle/freeware/
The free version lets you make as large a schematic as you want, but restricts the size of the board you can layout to a small area.
There are programs like DIPtrace which have 30 day trials, but after that period of time you have to pay, which doesn't seem like something you want to do so...
After a quick search on google of "Open Source Circuit Layout Programs" and I found,
http://opencircuitdesign.com/
This seems to be geared towards machines running Linux so if that is your platform then great, but if you have windows then,
after another google search I came across,
http://www.circuitstoday.com/pcb-design-and-layout-software
This site outlines a good chunk of the free open source layout programs out there.
I am not sure how much support or community these open source programs have around themselves. So if I were you I would get an Eagle licence.
Eagle has a strong backing in the electronics community, and a wealth of existing designs you can just load up a pre-existing design to get a hang of the program.
A hobby licence is about $170 US dollars and gets you a layout size of around 6 inches by 4 inches to play with and you can make up to a 6 layer board.
The free licence gives you about 4 inches by 3 inches to play with, and gives you up to a 2 layer board. The free license seems to be fitting for a beginner however especially if you are going to fabricate these boards at home using something like the "Toner Transfer" method. Even if you are sending these boards out for fabrication, the 4in by 3in size seems to be like the first tier on most fab sites in pricing.