If it's any help, I played three finger for about eight months and kind of got stuck where I wasn't improving like I thought I should. I tried clawhammer and in a very short time I could tell it was for me. I am improving at a much more rapid pace now. I wish I would have started playing at age 16, not only do I think young people learn faster but, I would now have 30 years experience. Enjoy the process of learning instead of focusing on an imaginary finish line. BTW I'm not trying to say that one style is better than another only that you will probably find one that is better for you if you keep at it.
Hey I just started on the banjo as well. Many factors affect the outcome of your question, ocms doms, such as natural aptitude, previous musical experience, age (you have this one on your side!) enthusiasm and available time, and so on. The banjo is a relatively easy instrument to get going on and I think someone with a good musical background, a natural ability to pick things up as well as buckets of enthusiasm and spare time could probably play reasonably well in a few months, enough to impress the untrained ear, but as with any instrument it can be taken to great heights with years of dedication and experience, playing with other musicians and listening to lots of different players and all sorts of music will all add to this. If you have never played an instrument before, and have only average aptitude for music and limited time to practice, expect any instrument to take a couple of years to get the basics down, and longer to really start to 'fly' when you play. So, although I have tried to give you a couple of examples of how long it can take to learn and instrument, you can easily see that the question is individual to you and the only person who could even attempt to give you a vague idea would be your teacher. He/ she can see where you're at musically now, and how easily you are picking it up, how quickly you're moving along and so on, and take a wild guess at where you should be in a year all else being equal.
I have to agree with Big Doug here, get into it! Enjoy your banjo time, spend lots of time listening to your favourite players and discovering more players, it's all about the journey. Those little goals that keep you itching to pick up the banjo, those little milestones you reach and immediately look ahead to the next one.... it would soon get boring if there was nothing left to strive for!
You are probably playing bluegrass style (sometimes callled 'Scruggs Style') banjo. Especially if you 'pull up' on the strings with the index and middle finger picks and keep your hand pretty still while you're playing.
Clawhammer is named that because the whole hand looks like it is knocking or 'hammering' on the head of the banjo. Up and down from the wrist. And the finger strikes downwards on the string, it doesn't pull up.
A lot of the material (the tunes and songs) might be the same, so you can't really tell that way. They are really different right-hand techniques for playing.
The first few years I had been playing, whenever I encountered someone whose playing moved me, I always asked them how long they had been playing. Inevitably they would say at least five years. That taught me that I was probably going to have to put in some time before other musicians would consider me good. My nonmusical friends thought I was good after about a year. The musicians who I greatly respect have been complementing me recently (about six years in) and that is an awsome feeling, but noone has asked me to play on their record yet, if you know what I mean. Someone once told me though, that if I was meant to be a faomus banjo player, I already would be, because the brilliance would be oozing out of me so strongly. Since I'm not already a brilliant, famous banjo player, he said that I should just take it for what it is - a fun hobby, and enjoy it. So that's what I do, and I try to play everyday, and I continue to get more comfortable playing with others, and I try to learn a new song everynow and then from youtube or some recording. Sometimes I sit down with a good fiddler and find out that my rendition of the song is missing some of the structure that their's has, other times I've found that I did pretty good. Six years ago, I couldn't even have come close.
Its generally accepted that people learn anything faster when theyre young like you. I started too late. Guitar at about 32, banjo about 15 yrs later. Its a tough slog, but the learning curve will go exponential, once you catch the bones of a tune youll get a buzz, then youll motivate and want more. Its the early months when you seem to be going nowhere when a lot give up and play football.Keep at it, its all good.
Banjo---------------Musical choice of the anti-social (quote, Eric Idle) If I ever get this thing in tune Im gonna get it welded.