I'm happy to accept and use conservation of energy when I'm solving problems at Uni, but I'm curious about it to. For all of my adult life, and most of my childhood I've been told this law must hold true, but not what it is based on.
On what basis do we trust Conservation of Energy?
Answer
Let me expand a bit on Manishearth's answer. There's an idea going back a long time called the principle of stationary action. See http://en.wikipedia.org/wiki/Principle_of_stationary_action for a description that isn't too mathematical. In the 18th and 19th centuries century the mathematicians Lagrange and Hamilton found ways of using this to describe mechanics. Then in the early 20th century the mathematician Emmy Noether discovered that in Lagrangian mechanics if a symmetry of the equations existed this meant there was a corresponding conservation law. As Manishearth says, one example of this is that time symmetry means that energy must be conserved.
Strictly speaking, the symmetry involved is "shift symmetry of time". This means that if I do an experiment, the time I do it doesn't matter so I'd get the same result tomorrow as I do today. If this is true Noether's theorem means that energy must be conserved.
Experimentally we find that repeating experiments does indeed give the same results, and we also find that everything observed so far obeys Lagrangian mechanics. This suggests that energy is indeed conserved. Strictly speaking this is an experimental observation not a proof, but few doubt that the principle applies as the universe would be a strange place if it didn't.
Wikipedia has lots of articles on Langrangian Mechanics and Noether's theorem, but they're a bit intimidating for the non-mathematician. If you're interested in knowing more Googling should find you plenty of more accessible articles.
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