I know how to use 比 in simple sentences like
哥哥比姐姐高 (my big brother is taller than my big sister)
Would it be used for other sentences like
You learn more words by listening than by reading.
you already have the answer! just remember when using 比, having an adjective at the end makes it very easy to build sentences, such as 高 in your first example.
building on what you already know, a more colloquial way of saying your second sentence could be:
you can also swap 快 with 有效 (effective) or 有效率 (efficient)。
You could simply translate "than" into "比" in Chinese, but in Chinese, there are several different patterns for comparison.
grace's answer is good, but I think his translation is more likely a liberal translation(I would like to translate that Chinese sentence into English word by word: Listening can enlarge your vocabulary more than reading .You see, the subject and object are different from the original sentence). By translating the English words into Chinese one by one, you get:
To keep the sentence structure the same, I would like to use the pattern below:
*I think you need a model verb, such as "may" or "can" in your original sentence, that's why I add "可以" here.
"哥哥比姐姐高" is correct. The next sentence "you learn more words by listening than by reading" can be translated like: