Your headwaters are formed by three things, although not always exclusively one:
- Snow Melt
- and/or an Artesian Spring (a good classification system is here)
Each has a unique geographic quality to them, and your story will have to procure the appropriate "source" of your river.
You already know that small creeks from runoff and rainwater form into larger and larger rivers until you have something like the mighty Columbia river.
There are four basic types of river:
- Young - think steep, relatively deep, and short; not a lot of historical erosion.
- Created - caused by a tectonic event; these are around more than you might think. This will give you your grand canyons, although not exclusively.
- Mature - long, with lots of tributaries and picks up a lot of sediment along the way to dump into the sea, such as the mighty Columbia and Mississippi rivers.
- Old - a long, flat, wide river with low erosion left, such as the Nile.
Waterfalls & Diversions
The basics of geology, insofar as water processes, are your three types of rock:
- Igneous (very hard, granite, etc.)
- Metamorphic (like gneiss and marble)
- Sedimentary (sandstone; "soft" and easily breakable).
When an igneous rock is formed from lava - note: I'm being very elementary here, I know - it pushes up through other rock types and can form mineral dykes or outcroppings of very hard stone.
This means that a water body can erode the soft stone, leaving the hard stone. This is exactly what happens when a waterfall forms, for example. The softer stone was eroded away, but not the harder granite, so the water has to go over the hard stone and drop away where there once was soft stone long eroded away. Diversions happen this way too.
Another forming factor in flatlands is the soil and sediment, which can cause a river to have the long, sinewy look in very fertile parts of your world. A once-straight river pushes sediment aside, which builds up. The side that has the most becomes a barrier, so the river goes the other direction; subsequently, the faster moving (furthest) side builds up. This creates a S-curve as well as oxbow lakes.
At the culmination of a river into a sea, lake or (more likely) ocean, a lot of really fascinating things happen. A mature or old river has long flattened out the end of its geology, and fans out into a delta. Because of the sediments it has brought with it, lush flora typically is present, such as that at the end of the Nile.
This also brings a lot of hungry fauna, including people. Furthering the biodiversity is the mixing of fresh and salt waters in the estuary. This is where I go fishing!
I shouldn't have to stress how important rivers are, and at the risk of starting a list:
- Transportation of Goods
- Water Supply (on mainland and islands)
- Energy Generation & Engineering
- Agriculture (leading to mythology about fertility)
- Hunting & Fishing,
... and on, and on.
Bonus: Have fun!
It is perfectly plausible to have in your story a navigable underground river, super short river, or a river whose watershed is bigger than Alaska. I should think rivers of different material composition on your world (rivers and lakes of methane?) would have a different impact on life on your Earth 2.o.