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Don't jump into the deep end
When picking something to work on, don't be the sole person to take on a huge task as your first contribution. Picking a task that's too large significantly raises the chances of failure. Also don't pick several things on several teams to work on. Start small, picking at most one or two things, and grow from there. The key is slow, steady, and sustainable growth. Don't join with the immediate goal of becoming the next leader of the project. Start small.
First contact
After you've decided what you're looking to do and what team you are looking to do it with, it's time to send an introduction to the list. When sending an introduction (usually by mail list), include the following information:
Name
Time Zone / Country
Basic skills and experiences
Why you're joining
What you're looking to do (be specific)
How much time you can contribute (usually hours per week)
If any of the above questions are not clearly answered, don't send the email yet. You're not ready. Remember, be specific about what type of work you're looking to do. Saying "Whatever needs to get done" isn't helping anyone. Saying "I'd like to help document system A," "I'd like to translate software for my native language," or "I noticed this webapp is particularly slow sometimes and I'd like to help fix that" is perfect.
Find a mentor or sponsor
This step is both incredibly difficult and important. Finding a proper sponsor will increase your chances of being a successful contributor significantly. Sometimes it's absolutely required. A sponsor will help with training, introductions and teaching new contributors how a team works.
Most teams have mailing lists. Email the list, say you're looking for a sponsor, and explain what you are wanting to do. If you haven't heard back in a few days, reply saying that you are still looking. ''Keep doing this.'' Most sponsors are people that have been in the project for a long time, and are often very busy.
They don't mean to be rude and don't want to send the impression they don't want new contributors. It's just that at the moment, some people will assume other people will take care of you and so for the moment, no one does. This is a common problem -- in real life as well as in online communities -- and a difficult one to fix. But sticking to it and continuing to ask for help without being annoying will show that you are serious and ready to contribute. Don't send this kind of message more than once every couple of days, but be positive, and persistent if needed.
Contributing
Once you've got something to work on, it's time to actually do work. The first several tasks you will work on will likely be small or maybe mundane. Do them consistently, conscientiously and well. This will raise the level of trust you have from the other team members.
As with other volunteer organizations, there are high turnover rates in the free software universe. Training volunteers is time consuming, especially for more complex tasks, and requires a commitment from currently busy volunteers. Spending days or weeks training someone only for them to vanish can be disheartening for mentors and sponsors. By giving out small tasks that have been hanging around, a sponsor can help you take small but vital steps, and learn whether or not the work you're going to be doing is really for you.
Look for work
If you have access to a repository, system, or content, consider yourself a partial owner. This doesn't mean you should immediately re-design everything. Remember that other owners have time and effort invested in the current material as well. It does mean, though, that you should take pride in the work you are doing. If you see something not quite right, do research on it and notify the list. Seek work out, keep yourself busy and help others.