How long will it take me to become good?

The answer to this question depends mainly on yourself and how much time you can devote to practice.

Obviously the more you practice, the faster you will improve. Assuming you practice regularly and have a lesson once a week, you should be able to play something musical in a month to six weeks, maybe less. Having said this, everyone is different and some people will pick things up more quickly than others due to a natural aptitude. It is never a good idea to measure your own progress against that of others. Learning a musical instrument is not a competition or a race. In the end the only opponent is yourself.