Testing the mobile-friendliness of your site will give you a solid jumping-off point for remedying various issues Google identifies. There are a multitude of metrics Google will test: speed, size, structure, etc., so working on those alone can carry your site well into the age of mobile. Click here to get started.
AMP is an open source Google initiative that seeks to remedy usability of websites on mobile devices by serving up content in a fast, minimalist fashion. Luckily, AMP pages are super-easy to create, so you can take advantage of Google increasingly showing AMP content (images included) in search results. In regards to SEO, the working theory is that Google gives bonus points to mobile-friendly sites leveraging AMP content.
This style of web design puts devices and platforms at the forefront such that a website's content will be displayed in a readable and functional fashion no matter the visitor's device screen size, orientation, etc. It's also Google's preferred style of web design, so if satisfying Google is your golden standard (it's not a bad golden standard to have), consider RWD as a way to remedy any mobile-friendly structural shortcomings your site may have.
When focusing on the markup and code of your site for SEO benefits, not all browsers are made equal. Luckily, there are sites out there to help you equalize the playing field, so you can drastically minimize the headache of making your site look coherent across Chrome, Firefox, Safari, Edge, etc. Yes, it's a pain, but a necessary one to provision for your site performing as you expect it to.
The all-too-common saying is that content is king. While content IS important, the key takeaway is to do content the right way, which is to make it flow as naturally and often as possible. In other words, do identify keywords and facets of related context (analogies, references, etc.), but do not overindulge in keyword distribution or repeat the same thing 80 different ways. Be smart and targeted, but do so in a natural context and keep giving Google reasons to crawl your page with fresh content.
Machine learning and AI aren't just the future of Google; they're here now. You can read about RankBrain in detail here, but the big shift to SEO is how RankBrain enhances Google's algorithm, improving how it returns results for vague queries posed in the form of natural, long-tail questions, much like you would ask a person a question. The key inference many SEO professionals draw from this is to build out individual pages that focus on a topic, not just a keyword or three. This stands to mark one of the biggest shifts to an age-old SEO staple: Highly-targeted pages based on laser-specific keywords alone.
The mobile-friendly evolution of rich snippets is rich cards. Basically, it's your content in Google served up as large (relatively speaking) images that really draw the user in. Implementing rich cards (via schema.org structured markup) plays directly into a mobile-first SEO strategy.
Without delving into the technicalities of Progressive Web Apps, they're essentially a new design paradigm that allows you to not have to deploy mobile app versions of your site to the App Store, Google Play, etc. Mobile users coming to your site can be served functionality directly from your site that you would otherwise have to have in the form of a phone app. The "indexable" aspect of it is where we care insofar as SEO implications, but the technical waters get deep quickly. As such, I'll defer to Google to elaborate for you.
Culminating a list of excellent resources to help you keep your finger on the pulse of SEO can really bolster your efforts. Here's a small sampling of my favorites to keep up with, all of which I highly encourage you to spend some time on: Moz, Seer Interactive, Search Engine Land, Search Engine Watch, and Search Engine Roundtable. There are countless amazing SEO sites out there these days, but those five (and ZDNet, of course) will adequately keep you abreast of the very latest.
If you're just getting the ball rolling with SEO, there are plenty of excellent resources out there to get you started. The SEO landscape is changing at a very rapid pace right now, so it's more important than ever to get the ball rolling and stay up to date with what Google is up to (as well as other search engines, should they become notably competitive). Remember, mobile first in 2017.