PCB design is not something you can start with quickly. First thing you need is an EDA (Electronic Design Automation) package, that's your CAD. You use the EDA to design your schematics (which for a breadboard you may have done with paper and pencil) and convert them into a PCB layout. EDAs often have their proprietary file formats, but for delivering your work to a PCB shop there's a standard format: Gerber, which your EDA should be able to produce. The Gerber files (one for each layer) together with drill information allow the shop to produce your board.
Creating a board layout you have to work with DRCs (Design Rule Check), which checks if you're not violating certain rules, like components or traces too close too each other. Your EDA will have a standard set of design rules, but you should be able to change them to your needs. This requires some experience, just like creating a layout in the first place. If you start with a rather complex design (many components, several of the with high pin count) it's very likely that you get stuck just trying to place the parts on the board so that the nets are routable. Many design companies have PCB design engineers who're doing just that. That's because experience is important and with experience comes intuition. An experienced layout designer may do a nearly optimal component placement from the first time, while someone with less experience may have to start all over again a few times.
Eagle seems to be a popular EDA software around here, but I haven't used it myself, so I can't comment on its ease of use/learning curve. There's a free version with limitations; IIRC your board is limited to 80mm x 60mm.