Before you start contributing to WordPress core, you must go through the Core Contributor Handbook.
Following are the ways to contribute to WordPress Core.
Go through the Contribute with Code chapter and its sub-chapters to learn about committers, component maintainers, SVN & Git repositories, the bug tracker (Trac), submitting tickets & patches, and much more.
The article, A Developer’s Guide to Contributing to WordPress Core, gives you information on where to find tickets, setting up a development site, running unit tests, creating a patch, and much more.
If you do not know about Subversion (SVN), then the Using Subversion document will help you to learn.
From the WordPress Core Components page, you can find a list of components along with their maintainers. You can also find existing tickets by components.
Follow best practices while contributing with code. It helps to keep WordPress code clean and well documented.
Tutorials & Guides will help you to find required information, or you can find your answers.
Go through the Contribute with Testing chapter and its sub-chapters to learn about several types of testing, reporting bugs including security issues, testing patches submitted by developers, and much more.