[subject] + 沿着 / 顺着 + [place] + [verb]
沿着 and 顺着 mean "along". The path – a road, a river, a wall – comes right after them, and the whole phrase goes before the verb of motion.
沿着 follows an edge or a line beside you: a river bank, a coast, a wall. 顺着 follows the direction something runs or points in, and can take a sound, a finger, a smell.
For a road, either works.
Both carry over to plans and ideas in written Chinese: 沿着这个方向 "along these lines". 顺着 with a person means going along with them: 顺着他说 is "say what he wants to hear".
English puts "along the river" after the verb. The Chinese phrase goes in front of it.