Though English has words like undergo, ongoing or rearrange, making new words from old ones is not a particularly productive way of building the core vocabulary in the English language. Russian has quite a number of prefixes and suffixes routinely used in words, including those in your essential vocabulary.
Such verbs have natural polysemy (in layman's terms: a handful of somewhat related meanings). For example, собрать means both "to gather, to pick up" (eg. flowers) and "to assemble" (eg. an IKEA chair). Both meanings grow from the parts that comprise the word: брать is "to take, to pick", and со- adds a meaning of "togetherness".
It is important to understand that only a few prefixes are like «пере», i.e. have a very focused meaning. Most Russian prefixes behave similarly to English prepositions when you add them to English verbs as particles. One's knowledge of English helps one guess what "turn up", "take off" or "run out" may mean. However, you can never be sure without a dictionary or a context that makes the meaning obvious (or you can ask someone to explain the word to you).