Eagle routing (manual routing)
This is usually the procedure I use:


Create your desired board outline. If not defined by an enclosure, you may need to tweak it later on, but start with what you think you'll need. This would also be the time to determine how many layers you need and configure accordingly.
In the PCB layout editor, begin by placing your connectors and other human-interface items (buttons, switches, LEDs, etc). Usually you want to place these in a specific location on the board, to make it neat and clean (in a logical manner for human interface).
Mentally group together different parts of the schematic - Power, Microcontroller, Communication, Logic, etc. Pick which one you want to start with.
In the PCB layout editor, drag the components which are a part of the section you chose to start with and group them together (outside the board but separate from the other parts). You may have to break the larger section up into smaller groups, depending on the size. For example, the Power section of the schematic may have several voltage regulators, so do one regulator with its requisite circuitry as one group.
Within the single small group you just moved, start positioning the components in a logical manner that promotes the easiest routing (shortest, straightest airwires).
Within that single group, you may start routing the connections. Don't worry about the connections that go to different sections yet, just route what will be self-contained within that group.
Move the group onto the board and find a logical place for it.
Repeat steps 4-7 with the next group of components in the section, and then with the remaining sections. 
Once all groups have been moved to the board and positioned where you want them (make sure to group parts within the Power section together, and parts within the microcontroller section together, and so on), you can start routing between the sections and do the human interface parts.
During the routing process you may find it much easier to place copper pours instead of individual traces, usually for power and ground connections. I normally do this right after I place all component sections on the board. This will minimize the airwires for ground or power that are visible and make it easier to identify which connections go group-to-group.


Sometimes you may need to do some of these steps out of order, but you'll get a better feel for it the more you practice. However, the process mentioned above should help you get started with routing just about any board.