First of all, kudos to you for thinking about contributing to bigger projects, and open source ones at that!
Projects with large codebases always take some time to figure out. Therefore, it's wise to start with a bug that a more experienced programmer has labeled "bitesize". Other commenters here mentioned Josh Matthews' super-cool "Bugs Ahoy" tool for Mozilla -- http://www.joshmatthews.net/bugsahoy/. I'll chime in that I'm a maintainer of a similar thing that covers even more projects: https://openhatch.org/search/
If you want a general roadmap of how this works, we (OpenHatch) have also written up a step-by-step guide here: http://labs.openhatch.org/foss-contrib-guide/ . Work through it as a checklist, clicking the box next to each step as you do them. We've found that works really well with people.
The two big things in my opinion, as you start out, are:
* Find a project that you care about. Otherwise, you'll lose momentum. It's cool that you want to work on Mozilla-y projects! My only warning is that the long build time means you are going to have to plan to wait a while (about 1 hour) between downloading the code and actually running your local build.
* Find a task that is doable, so you can actually get it fixed, and you actually enjoy it!
Some things to watch out for:
* The build process could be super slow, for big projects like Firefox.
* Eventually you're going to have to get used to navigating enormous projects. This might sound daunting, but actually I think you'll find it fun after a bit.
* Make sure you have a place where you can reach out and get help. You'll probably be running into all sorts of new things -- version control tools, patch submission guidelines, an enormous codebase -- and you're more likely going to have a good time if you can get your questions answered.
For that last one, feel mega free to join #openhatch via IRC on freenode.net (more info on that here if you don't know IRC -- https://openhatch.org/wiki/Open_Source_Comes_to_Campus/JHU/L... ). OpenHatch is a community of people welcoming to newcomers in open source generally, and we can usually help you straighten out whatever problems you're running into.
There's also #newbies on irc.mozilla.org; I'd recommend joining there, too.
If you want a quick tour of what it's like to submit a patch, you can work through the OpenHatch training missions at https://openhatch.org/missions/.
You asked for a "flexible time framework" -- the good news, it's all on your own timeline. You can show up whenever you want. If you can explain more about what you wanted with that question, I'd be happy to explain.
Good luck! Welcome aboard!