The only practical definition of subordinate clauses is that they function as nouns, adjectives or adverbs in relation to another word or construction. Main clauses have no such function: their pattern is the pattern to which everything else in the sentence relates.

      In analyzing clause structure, most students find it easiest to work backwards, starting with the last S/V in the sentence. For example:
 

Find all the prepositional phrases and S/V/C patterns first:
 

Having found the last subject/verb ["we were going"], you next have to decide where the clause begins and ends, i.e. which words "go to" this subject and verb. The "or" joins the two prepositional phrases "on a picnic" and "to the mountains," both of which explain (and thus "go to") "were going." Thus the clause ends with the word "mountains."

At the other end, in front of the subject "we" sits the word "that." The question is "What is it doing there?" It does not mean "that we," i.e., it does not "go to" "we" in the sense that it goes to "house" in the sentence "I saw that house." Nor it is the direct object of "told": the sentence does NOT mean "Mom told me that," it means "Mom told me that we were going ...." The only other possibility is that this "that" functions as a subordinate conjunction. The "that" is thus the first word in the clause, and the clause is subordinate.

If the clause is subordinate, it should chunk to some word outside itself. Since the clause answers the question "Mom told me what?", it chunks to "told" as its direct object. We now have:
 

Having finished the "we were going" clause, we move backward to the next S/V/C pattern - "Mom told ..." Because the "we were going" clause is the complement of "told," that clause is part of the "Mom told ...." clause. Thus the last word in the "Mom told" clause is "mountains." To find the first word, we look in front of the subject, where we find the word "when." Since "when" can be a subordinate conjunction, we may have a subordinate clause. To verify that, we need to see if the clause "when ... mountains" chunks to anything outside itself. Since it indicates when "I became happy," it chunks to "became." Thus it is a subordinate clause, and we now have:
 

Students are usually amazed to find clauses within clauses, brackets within brackets. But so the language works. We next move to the "I became clause." Since a clause is a S/V/C pattern and everything that chunks to it, the "Mom told" clause is part of the "I became" clause." Thus the last word of the "I became" clause is "mountains." Since "happy" is the complement of "became," it too has to be part of this clause. In front of "happy" is the word "how." "How" can be a subordinate conjunction. Is it here? We can test it. Suppose that we consider it to be one. We would then have a clause beginning with "how" and ending with "mountains." Would that clause chunk to something outside itself? Yes, it would be the direct object of "I can remember." We now have:
 

Since we now have only one S/V/C pattern left, all we need to do is to put a vertical line at the end. 

     The preceding analysis may appear complicated, but with a little practice you will find that it is not that difficult. Besides, in analyzing texts, instead of the sentences in a grammar book, you will find that you already read and write sentences like this regularly. One of the major problems with most approaches to grammar is that they focus on teaching the constructions (subordinate clause) but generally ignore how the constructions are embedded into one another in actual writing.
 

This border is a reproduction of

 Master Hare
1788, oil on canvas, Musée du Louvre, Paris
Carol Gerten's Fine Art http://metalab.unc.edu/cgfa/

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