You're taking a girl on a date. You're trying to suggest fun things to do together. There's a pretty river near you, so you suggest:
Do you want to go for a stroll along the river?
This is one way to suggest an activity. Use this when you're being polite and not sure if the other person will agree.
If you're more confident, you can say:
Why don't we go for a stroll along the river?
To "go for a stroll" means to walk somewhere in a slow, relaxed way for enjoyment. People usually "go for a stroll" in a park, around their neighborhood, on the beach, etc.
Use the preposition "along" to describe walking next to a river. "Along" is also the word to use to talk about things that are next to a river in several locations:
There are some pretty yellow flowers growing along the river.
You can also use "down" or "up" to talk about walking next to a river. "Down" is the direction that the river is flowing in, and "up" is the direction that the water is coming from.
I walked up the river for about a mile, then turned around and walked back down.
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