Using SP 2013 on Premise and wondering how I can setup my sites to work on all screen resolutions? Right now everything looks good for me but on another lower resolution screen some zones are overlapping others and promoted links are being cut off and so on. Does anyone have a solution for this?
You can achieve this only by modifying the SharePoint master page. As I think you do not want to start from scratch you could use some pre created template.
MVP Sasisprite has created a responsive master page for SharePoint that works with SharePoint 2013 as well.
See this article for your reference: https://gallery.technet.microsoft.com/office/SharePoint-Branding-e73dffa0
Right way which is also out-of-the-box solution, is to use Team Foundation or Team sites which contain the modern publishing pages. What you might be using now are Webpart page or a wiki page, they can also be made mobile friendly through custom styling/scripting. For example you can use bootstrap to set custom Master page layout which will then auto align navigation bars and other elements of page. But publishing pages(modern page) are mobile friendly by default.