The technology industry moves at a fast pace, and the broad range of technologies used by full-stack developers means that they will need ongoing training.
The first step in staying up to date is understanding which new technologies and frameworks are gaining popularity. The best way to do so is to follow online technology blogs and communities, such as Hacker News or Stack Overflow. Github - the most popular host of open source code - publishes a list of up-and-coming projects.
Once you know what to focus on, the next step is to find good resources to learn from. There is a wealth of online information for languages and libraries, and most of these will have resources and getting started guides on their websites. Youtube is increasingly a good source of tutorials, and most technology conferences upload their talks and occasionally even stream them live on the site.
The most reliable source of training will come from online MOOCs - Massive Open Online Courses. These courses emulate a traditional university course, and in some cases can even lead to certification upon completion. Examples of companies that produce high-quality technical MOOCs include Treehouse and General Assembly.