You must wash out the equipment you're using with the substance you're putting in. An exception is the substance you're titrating for in the beaker, in which the important thing is the number of moles of that substance. For example, after if you want to transfer say 25mL of NaOH to the beaker for titration, it would not matter if you wash it with distilled water before hand, because the important thing is that there is 25mL equivalent moles of NaOH in that beaker. However for the burette you MUST wash it with the substance that's going in, or else it would modify the mole ratio of that substance vs whatever you washed it with in the burette.
A good idea is to put labels on all your beakers/flasks etc. so there is no mix up. Another good idea is to use a control, that is if you are looking for a colour change of a certain colour due to adification for example, it would be wise to set up such an acid with a bit of indicator in the first place before the titration so that you can easily match it up with your titration to spot the end point.
Another error is once you're around your end point, swirl the beaker for a good few moments before adding another drop. Sometimes the substance takes a few seconds before even starting to change to another colour.