This is a question a I got a lot. Today I saw the question What's the best way to learn device driver development on Linux? on Quora so I spent a few minutes answering it.
The answer finally was longer than I thought so I decide that it was worth a post. I don't know if this is the best approach to learn Linux kernel development and get involved, but is the one I recommend nowadays.
I for example didn't follow these steps, I got involved first watching Greg KH's Write and Submit your first Linux kernel Patch excellent video. After that I realized that anyone (even me) could contribute to the Linux kernel so I cloned the linux-next integration tree, compiled and started sending trivial one-liners patches.
But, after getting a job as a Linux Kernel Engineer and being contributing to the kernel for some time now I realized that there is a better and more fun way to get involved. Obviously device drivers is the best way to get involved for two reasons.
First, unless you are a PhD student doing research on operating systems. It is very unlikely that you will came with a brilliant idea an start hacking the Linux kernel virtual memory or block device subsystem. The Linux kernel has 20 years of development and its core is very, very complex. Second, it is more probable that your first job as a Linux kernel developer will be to write device drivers.
But, whatever is the way you chose to get involved, there is a step you can't skip and is to learn the theory behind operating systems in general and the Linux kernel in particular. So if you don't know about operating systems, I recommend you to read Willam Stalling's OS book. This book has a more hardware oriented approach unlike other OS books that focus more on the algorithms and data structures used in operating systems than explaining with examples how things work in the real world.
After having a high level overview of operating systems, you have to learn about Linux kernel development. Robert Love's Linux Kernel Development book is for me the best one on the subject. You can learn not only about Linux kernel development, but also the motivations behind the technical decisions that lead to the design.
Then you have to learn about device drivers development on Linux, two books that are very good are Linux Device Drivers and Essential Linux Device Drivers. This two books are complementary. The former teaches how to write virtual device drivers for memory based devices, so you can try the examples without the need of special hardware. While the latter shows you how device drives for real devices are actually written. I maintain an up-to-date repository with all LDD3 examples so you can compile and test on recent kernels.
Once you have the theory you can begin with the practice by writing real device drivers. You need a hardware that still is not supported on Linux. The good news is that you probably already have on of this devices.
With the popularity of the Android platform, is very likely that you own an Android device. Even when Android is a Linux-based operating system, Google by a design decision forked the Linux kernel and added some APIs that don't exist on the Linux kernel. These APIs are used by device drivers and for that reason, a developer has to choose whether to write a device driver for Android or Linux.
So, porting Android device drivers to Linux is an excellent opportunity to learn device driver development skills, the Linux kernel development process and how to work with the Linux community while doing something useful.
A final comment, when asking questions to kernel developers on public mailing lists, remember to do your homework and researched first. They tend to be very polite and collaborative with people that show signs that they tried to understand the problem first before to ask, but can be a little grumpy with people that ask first before even tried to understand the problem.
Also, don't be afraid to contact the developers that wrote the Android version device driver. They usually are working for the company that build the hardware of were contracted by them. Usually both the company and the original developer also want the driver to be merged on the Linux kernel but they don't have neither the time nor the resources to do it.
So, if you offer them to help and your time to modify the driver to be in an merge-able state, they will be more than happy to help you finish the task. This is important because you probably will need the hardware datasheet or at least information about the people that know well the hardware.