In my opinion, the first thing you should do is write down a list of things that you're really passionate about. Is it languages? Then find a language based repository on GitHub/GitLab. Is it networking? Contribute to a project that interfaces with HTTP.
"First of all, how do you pick an area to contribute?"
I can see from your submissions history that you are interested in AI and some machine learning. Those two somewhat related fields are fields that are hot right now so finding projects on them will not be difficult. You don't need to join a super large or complex project, it could be as simple as contributing to a series of tutorials on machine learning.
"Do you go with something that's related to your field of expertise?"
Do you want to learn something new? Then find something that's not related to your field. Otherwise if you want to solve a problem in your field of expertise, then sure go with your area of expertise.
"Is it even feasible to instead pick an area that you want to learn more about?"
Yes, but it depends on the area. If it's developing an operating system, then be prepared to do a lot of reading beforehand before you even think about contributing. Project maintainers who are in charge of behemoth sized projects (such as those in charge of maintaining an operating system) tend to have little to no time to field questions. Whereas small projects and their maintainers have ample time and, in fact, would love to answer questions about their projects.
"Do you jump straight in and start fixing bugs to get started?"
Personally, yes I have if it solved a particular irk I had. Although, I would first open an issue and detail what you plan to contribute. Then send a pull request.
"What's the typical timeline by which you feel like you are being productive?"
Typically, a few months in I became really productive.
I would also suggest creating your own project rather than contributing to it. But be selfless and open-minded with it. Allow contributions and let people criticise it - it's all part of the learning experience. Helping other devs never feels like a chore and actually ends up being enjoyable.
The best thing about open-source is that I felt like I was contributing to something bigger than me. That my projects can and may live on long after I've gone. When you receive your first issue or pull request it gives you a sense of happiness that people are using something that you created. For me, that is one of the best feelings ever.