The most important thing is to first determine if it is "wear-and-tear" arthritis (osteoarthritis), or if it is an inflammatory arthritis. Most of the time a GP or primary care doctor can do this for you.
If there is sudden swelling and redness, then it is less likely to be plain arthritis. With osteoarthritis the bones of the joints will enlarge and look misshapen, but it is obvious that they are firm bone. There is usually not any redness. Movement or firm touch can cause some pain.
With inflammation the enlargement is softer and usually red, and the joint is more tender to the touch. The joint will be warmer than the surrounding akin. There are many causes of joint inflammation, ranging from minor problems to more severe. A few blood tests and xrays should be able to tell your doctor more.
The younger you are the more likely it is for an inflammatory arthritis to occur. After a certain age (40+) these diseases do not often begin and it is more likely to be regular arthritis. Gout is also common in either age group, but it rarely shows up first in the fingers.
Give your doctor another chance to figure this out. Ask for screening blood tests and xrays looking for inflammation and gout, and be sure to remind him or her of how important your hands are. It's better to have a diagnosis before picking up medication to try...
Good luck!