Answer: The sentence contains a single independent clause.
Explanation: There is only one subject+predicate pair in this sentence: "We" (subject pronoun) forms the whole subject, while the predicate is formed by "took" (verb) + "a long walk" (main noun phrase of the object taken by the verb took) + "along the path" (prepositional phrase modifying walk) + "beside the bubbling stream" (prepositional phrase modifying path). There are no other verbs, therefore this is a simple sentence (simple sentences always consist of a single independent clause).
