How best can a product marketer keep up-to-date on their skillset or formal learning in a world where marketing tools are constantly becoming obsolete?

First, remember there is a difference between a product marketing and campaigns/demand generation lead. Product Marketers focus their time on customer research, value prop, go-to-market plan, competitive landscape, and market programs. In companies big and small, you would probably have a demand gen specialist who would keep up to date on marketing tools in order to optimize spend across channels. 

If you are part of the team that is investing in marketing technologies, you should talk to your customer, refresh your personas, and track changes in preferences over time. As a product marketer, your ultimate job is to know the customer better than anyone else. Identify what channel your customers want to be communicated through and then investigate technologies that would help you optimize your outreach. 

I keep abreast of innovations in marketing technologies by networking, attending conferences, or coffee chats with mentors. When talking with peers, I usually ask what's the most impactful book or blog they read that helped them in their careers. I'm also a huge believer in having mentors that help navigate what technologies companies are investing in. When I notice a trend, I'll research and start learning the tool as a side project.

I struggle with this too. The best way is to keep in touch with the PMM community through Sharebird and PMA slack channels. Outside of that, I am on the lookout for any relevant courses or certifications or conferences to attend. I also have membership to medium, a platform where professionals share their experiences. If you know any more active channels, please do share. Thanks