Taught me or showed me?

Solution 1:

They can both both be correct depending on the meaning you're trying to convey.

If you say the "experience has taught", then you're saying that you learned something of value from the experience. In this case you learned the importance of waiting.

If you say the "experience showed" then you witnessed something and may or may not have learned from it. For example, it's entirely possible to watch a presentation or a demonstration and take nothing from it.

However, in the given sample sentence, there is the implication that in being shown the experience, you did learn from it. So essentially in this example the two sentences are equivalent.