「行{い}く」 is a verb as you know. Here, the "act of going someplace" is the object that the speaker's mother found bothersome.
Because 「行く」 is a verb, however, you cannot say 「行くを」 as 「を」 must always be placed directly following a noun.
Thus, you need to turn 「行く」 into a noun form. How is that done? You can nominalize a verb by attaching a 「の」 or 「こと」 directly to a verb, which is like adding "ing" to a verb in English to make it a noun. This usage of 「の」 is very useful just as "ing" is useful in English. Frankly, you could not say anything intelligible in Japanese without using this nominalizer 「の」.
Both 「行くの」 and 「行くこと」 are in the noun form (both mean "going" or "the act of going"); therefore, you can now safely and grammatically attach an 「を」 to form a direct object of the verb phrase 「面倒{めんどう}くさがる」 ("to find bothersome").
Finally, your translation is okay except for the "taking me to" part. 「迎{むか}えに行く」 means "to go and pick (someone) up" instead. In addition, 「言うまでもなく」 means "needless to say".