USGS has a web map called Streamer that will let you trace a stream or river in the US. If you click on a point on a river, it will highlight every part of the river that's downstream of that point. Or you can choose the "trace upstream" option and it will highlight everything upstream of that point. For example, here's everything upstream of Hoover Dam:
Aside from a map such as that one, there are some visual clues you can look for on a normal map.
Dams and Reservoirs - if there's a dam, there's usually a reservoir upstream of the dam. So if your map shows a dam and reservoir, you immediately know which way is upstream on that river. Even if the dams aren't marked on the map, sometimes you can tell where they are by the shape of the reservoir edge. In a graphic map, a dam creates an unnaturally straight line with a wide stretch water along one side of it and a narrow strip of water plus bare land on the other side. Of course in satellite view you can see the physical structure of the dam itself.
Unfortunately you may not be able to tell from the shape of the reservoir which end is the outlet and which end(s) is/are the inlet(s). For example, Lake Mead, the reservoir above the Hoover dam, looks like it has a dam on each of its three corners. You would have to zoom in on each corner to see if there's a dam there or not.
Confluences - a confluence is the point where two or more streams flow together. Streams usually flow together at an acute angle. If you draw the acute angle on the map, it makes an arrow that points downstream. This is not 100% reliable, but it's helpful nonetheless.
The "Rule of Vs" - on a contour map, contour lines that cross a river will bend into a V shape, with the V pointing upstream (the opposite of confluence angles).
Find one end of the river - trace the river until it either flows into a lake or sea, or until it peters out. When it gets too narrow to see on the map, that's (approximately) where the headwaters are. In the case of major rivers like the Colorado, you have to trace the river for quite a long way.