I read in my Latin to English and English to Latin dictionary that the genitive plural of res is used to strengthen an adjective. However, my latin teacher said that he thought that if a superlative wasn't strong enough one could use valde in addition. So which is more correct, and why would rerum be used to strengthen an adjective?
I am not perfectly sure I interpret your question correctly, but I assume you mean one of two things. In both of them it is not that important that it is rerum; it could be almost any other genitive that fits the context.
I think rerum strengthens an adjective in a different way than superlative or valde does. For example, peritus means "experienced". Therefore "very experienced" could be valde peritus or peritissimus. If you want to say "experienced in many things", you could say variarum rerum peritus. This of course requires that the adjective can take a genitive attribute in the first place; for example peritus, cupidus, memor and plenus do so.
If you mean rerum strengthening superlative, then it is an instance of partitive genitive. For example rerum dulcissimus means "the sweetest of things". Similarly nostrum sapientissimus means "the smartest one of us".
I have to disagree with your teacher — if I understand correctly. I have never seen valde strengthening a superlative, and it sounds weird to me. Compare this to English, where it makes sense to strengthen "the most beautiful" to "the most beautiful of all flowers" but not to "the very most beautiful".