One of the myths floating out there we're hearing is "Hey, my website's responsive, so that's all I need to do to take advantage of Google's change in their index structure. " That's simply not the case.
Google looks at a lot more other than just whether your website responds well on mobile devices. You want to look at the structured data that's present in the code of your website. Structured data tells Google what that page on the website is talking about and will differ across different pages on your website. Take a look at that and make sure it's up to date, current and relevant to the people you're trying to draw to your website.
The next thing you want to review is meta data. Review the meta tags on your website to make sure they're still relevant as this is also an item that Google uses to understand what your page is talking about.
User experience is the next thing you want to look at. This is how users are experiencing your website. What they go through, during their journey through your website. You can learn a lot about what's working and not working by looking at your Google Analytics reports and focussing on mobile website users to identify certain pages or events that are making them drop off your website. This will present you with some opportunities to fix issues on your website so that mobile users have a great experience.
The last thing you want to look at is speed. How fast is your website loading for mobile users? Not everyone has a high speed connection available on their mobile device, so you want to make sure your website is built in a manner so that page speed is a priority and website pages are loading relatively quickly. I encourage you to review these items with your website to maximize your website's search engine position.