Emphasise correctness over speed until you are near perfect. Be super critical of your working and of your proofs, think of every possible place where things could go wrong and be painstakingly thorough in ensuring that they don't.
Obviously this style of solving problems will be very slow to start with, but once you have these rock-solid fundamentals, then you can seriously pick up the pace. More importantly, you can speed up without getting careless, as by this stage you will get a pretty good sense for when a solution is correct and when it is not.