Before you start contributing to WordPress Core, you must go through Core Contributor Handbook.
Following are the ways to contribute to WordPress Core.
Go through Contribute with Code chapter and its sub-chapters to learn about committers, component maintainers, SVN & Git repositories, bug tracker (Trac), submitting tickets & patches and much more.
A Developer’s Guide to Contributing to WordPress Core article give you information on where to find tickets, setting up development site, running unit tests, creating patch and much more.
If you do not know about Subversion (SVN) then Using Subversion document will help you to learn.
From WordPress Core Components page, you can find list of components along with 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 Contribute with Testing chapter and its sub-chapters to learn about several types of testing, reporting bugs including security issues, testing patch submitted by developer and much more.