I guess from the description that you have a flat table and don't want to make dig-in terrain. That forces you to have a very thin base. I would personally not go for card stock as it will most likely warp. Plasticard or similar might be better suited.
The river sections that i am using are dig-in terrain and have a more solid base.
To get the river look like a real river there are some specialized products available on the market that provide a very glossy surface. For my river sections I used an airbrush and standard acrylic paints to get a smooth gradient from brownish to very dark blue at the center of the river. Then I used Artificial Water as top layer to get the glossy river effect (
http://www.unrealdetails.com/fun-with-the-kids). I used artificial water to get a really thick coat and additional depth. When using build-on rivers then Woodland Scenics Water effects might be better as you won't pour on a thick layer anyhow.
The river itself is 10cm wide which is sufficiently large yet still practical.
Before creating the river sections I first experimented with making small lakes. It's less work so is mentally easier to throw away when you don't like it. So that might be a interesting idea for you as well...
Good luck
Jo