Whether your goal is to connect with living descendants of a common ancestor, or simply to continue to trace your heritage back further generations in time, you have two basic options. Find the people yourself, or have the people (or their relatives) find you.
The first option, finding the people yourself, is the most traditional. You search for records. You hire somebody who is on the ground in your locality of interest to get the records you need. You look at the records, transcribe them, translate them, compare them with other records, correlate the information found within. You cite with diligence every record you find so that others (including your future self!) can retrace your research. You write what you have discovered - you compile the information into an understandable format. These principles are the basics of the Genealogical Proof Standard.
The second option, having others find you, has only been available recently with advances in both internet and DNA sequencing technologies. The basic idea is you first find some information about your ancestor. Then, you publish it where others will find it, which is probably online. If you really want to facilitate speedy connections, you will join some sort of cloud-based family tree program.
For example, Familysearch Family Tree is basically a cloud-based family tree of the whole world. I have met dozens and dozens of people who are researching the same people. We are able to combine forces and research more efficiently and effectively.
Sometimes the person who you find has access to records that would otherwise have been extremely difficult for you to get. Sometimes they already have the information you needed. Often times, you both can share information. You both profit from collaborating. Accuracy of conclusions improves.
This kind of collaboration gives you a better sense of how connected we are in the human family. It is crazy to think that somebody thousands and thousands of miles away from me, who speaks mainly Czech, is from the same family; We have the same great great great great great grandparents. I love this feeling of connectedness. It is one of the reasons why genealogy is so addicting! Solving the puzzle and making connections, especially to living people, is extremely satisfying. It helps affirm something about my own identity.