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I would suggest you look into the Linear Referencing tools.
Using these tools you will be able to assign measures to each of the rivers/streams.
I would like to calculate_river_miles from the river mouth to certain stations on the river upstream. I can get river shape files online, but do not know how to calculate the distance to the station.
My approach is:
1) get the river vector files, and connect segments that are disconnected
2) designate the river_mouth (point where river dumps in the ocean) manually
3) if the station is not on the river, create a buffer where the point will intersect with the river
4) create a query on attribute table where the distance from river_mouth to the station upstream will be calculated.
5) some of these stations upstream will be on tributaries too, not just on the mainstream section of river.
Any suggestions to work thru the 5 steps would be greatly appreciated.