Here is a picture of red oak. The pencil points to the grain which shows as short brown hash marks, it's not the same as the annual growth rings that many folks call the grain. In the pic, the grain runs opposite the growth rings. If you were feeding this piece into a router bit, you would feed the end towards the pencil eraser first. An easy way to know grain direction is to cut a sliver off the edge with a utility knife. In one direction the knife will grab and chip out, in the other direction it should cut a continuous sliver.
Grain is more easily seen by sighting down the length of the board. Some species are easier to see than others. Often, boards may have grain going in two directions. Figured lumber like birds eye maple have grain going 360 around each eye. When laying out your pieces for a project, you try to identify the prevailing grain and think ahead as to how each piece will be run on cutting tools, not only routers but jointers, planers, hand planes, etc. Cutters need to be sharp. Don't be afraid to make multiple shallow passes, particularly with a router. It may take 5 or 6 incremental cuts to reduce/eliminate blow out, especially on species like oak that tend to splinter.