I have come to the opinion that regrets are the result of faulty perspectives. Every time I think I have come up with something I regret in my life I eventually realise I can't really call it a regret. My religious friends would describe it in terms of God's plan. I really have no idea if that is accurate. I guess I will find out soon enough. For the time being I will address more secular reasons for this belief. All of us have to assume some basic principals, whether we admit it or not. We don't like to admit it, but these principals are not rooted in logic and reasoning. They can't be: they are our axioms, our fundamental truths based on our accumulated experience, usually in the form of subconscious heuristics. I do not believe we can really know exactly why we believe these axioms. The best we can do is understand what they are and try to make them consistent. One of my principals is the value of learning. Books and tools of learning are in a sense sacred to me. Of course, this is due in large part to my upbringing. I hope this principal is more or less consistent with you own principals. For example, I feel the "God's plan" hypothesis is consistent with this because the Bible (and other religious texts) is littered with references to God being "the Truth" and so on. I also simply feel (yes, "feel") strongly that if there is a God, it has implanted a native need of learning in us.
Two important modes of learning are pleasure and pain. In fact, they are usually considered fundamental. Eventually every child gets too close to a fire and so experiences pain. We have a natural tendency to try to deny them that pain, but in so doing we would deny them the opportunity for learning. So if you let a child hurt himself (within reason, of course) should you regret it? If you hurt yourself, should you regret it? Even if you make a mistake that hurts someone else it is not entirely clear: both you and they may have learned something important. Hindsight is also an important factor. We should learn from history, not regret it.
Economists have a name for regrets: they call them "sunk costs." The idea is similar to "don't cry over spilt milk." We have a tendency to let our decisions be influenced by sunk costs. This tendency represents a logical fallacy. The classic example is buying a movie ticket and deciding you don't really want to go but feel you would be "wasting" the ticket if you don't. There is a saying popular in the game of Go (related to the better known games Othello/Reversi): Don't throw good stones after bad. That is, if a group of playing pieces is doomed, don't spend additional resources trying to save them. If something you have invested in doesn't work out, move on to something else, and, of course, find a way to learn from the experience. Future decisions should be for the future, not the past.