Both are easiest if you orient your map to North and check often that the map is still oriented. A compass may also be used to keep the map oriented. It's helpful, only if you trust the compass to be accurate. Most orienteering events use staggered starts to allow orienteers a chance to concentrate on their own course. There are several popular formats, including relays, mass starts, teams and recreational self starts.
Events take many forms as well.
Foot orienteering was described above. Others include using bikes, skis and even canoes. Orienteering is Fun.