Be sure that the strings are seated firmly in the holes when you're changing strings. Those little twangs are sometimes the string twisting in it's slot and changing the seat of the little roller end of the string in the string retainer.

Also, it sounds like your intonation is off on your guitar. Theoretically, a standard fretted guitar will never be in perfect intonation on all six strings at any given time. It's not physically possible. However, you should always be able to get it close enough to where people with normal ears can't tell the difference.

Search for "intonation" and see if you can find some spots where this has been explained a bit better... i know I have explained it at length on here before as well as a basic step by step of how to do it.

Hope that helps.