Can anybody help me with on-page SEO checklist and the biggest factors that affect my search engine rankings?

Yes, I can help you with a handy on-page SEO checklist that can give you the basic understanding of the major ranking factors ruling over there, as well as the standard ways of keeping everything in place with your overall optimization effort over there. Remember, the following on-page SEO factors listed below can give a huge positive impact on your website’s or blog’s general ranking potential in the relevant Google’s SERPS, of course when optimized properly. That being said, here is what makes a web page well-optimized for online search, as well as the main stuff that will need your most precise attention to that point – before anything else.

On-Page SEO Checklist – Basic Factors that influence Your Rankings

Taken in general, the whole concept of on-page SEO can be covered with a short list of main factors, simple suggestions, and practical ideas I’m going to outline below. Of course, there are actually much more different tricks, tricks, and techniques used by the best optimization experts – but the following primary categories of on-page optimization and the major factors mentioned with them are the very basics, which can become a nice starting point for every novice webmaster or active business website owner. So, let’s delve.

What Makes a Good User Experience

• Simple yet functional content structure to be easily "consumed" by every visitor, and well-optimized on-page navigation elements (such as menus, categories, CTAs, etc.) – everything should be intuitive and fully understandable for every online user on the first glance. 
• Clean & clear design of high quality, polished with a compelling, eye-catching look, and fitting your brand name concept.
• Sufficiently fast page rendering (i.e., up to 0.5 seconds to first response and no more than 3.0-3.5 seconds) to completely loaded primary content.
• Browser optimization & device responsiveness – make sure your web page content is running equally well on all top-popular web browsers, as well as displayed correctly and functionally for mobile browsing (i.e., fitting users who use smartphones/tablets for online surfing).
• The adequate page layout that clearly displays the most important elements of your content, with a well-thought visual structure of every separate web page/content section that makes scanning easy and without even single unnecessary step/click needed to reach any given section/category of the site or blog.

The the Right Way of Keyword Targeting

• Make sure to have one primary keyword or its long-tail search phrase appear in page URL, as well as the first words of Page Title. That way, don't forget to target the actual search intents of your potential visitors with each separate keyword. 
• Your on-page content structure should have a clear logic, posts/articles are to be comprehensive and fully informative (yet not too long or, God forbid, duplicated). Pay more attention to employing your main contextual keywords and related search terms in a natural way, without simply dropping a given number to maintain the keyword density (that old times have already passed long ago), not to say that simply dropping them at random is probably the best way of earning yourself a negative ranking penalty from Google search engine itself.

The Rest of Miscellaneous and Technical Stuff

• No dynamic parameters for URLs.
• Every content section or a given web page of your site or blog MUST be accessible with no more than four user clicks. 
• The right usage of 301-Redirects from other/outdated versions.
• Make sure your "robots.txt" file allows search robots for crawling (unless you purposely want to have something blocked from crawling & indexing).
• Consider the optimal length for the following major elements, such as Page Title (up to 75 characters recommended), ULTs (up to 90 characters recommended), Meta Description (up to 160 characters allowed).

• Don't forget to create a well-structured XML Sitemap, in line with including each URL over there – both for better indexing purposes (i.e., easy access for search crawling bots), as well as a clear logic order of human browsing throughout every section/category of your website or blog.
• Make sure to use your contextual keyword-based assets the right way: use one primary keyword prominently in the page heading, embed your body content with the keyword/long-tail search phrase to be mentioned at least twice and always organically, drop secondary keywords across the rest of your headings, Page Titles, and body text as appropriate (but make sure to keep a healthy balance between rich on-page content optimization for the main target keywords, and the natural user-oriented wording – remember that "keyword overstuffing" or "over optimization" is bad too), check your descriptive link anchor texts pointing to the pages within your clean & clear interlinking structure that must make sense for the live user first and foremost – but make sure not to overdo with the keyword-rich anchors and be even more careful with exact-match link text anchors (i.e., when done improperly or in a too vast quantity, they can potentially deliver much stronger negative effect on your overall SEO implementation, rather than bring any measurable benefits to your ranking progress over time).

Bottom Line – Uniqueness and Added Value Offered with Content

Ideally, your on-page content should provide only authentic (original) information, with a sufficient obvious value offered to every online user reading it – make sure to be more informative and descriptive with your content, in line with avoiding any explicitly stated self-promotion, irrelevant and blurred data which might as well be nothing but writing just for the sake of keyword optimization – that will never work. After all, you're creating your high-quality content for the live users before anything else, not for machine crawling and indexing, right?

Hint: simply ask yourself – whether you can know it for sure that your web pages would be described as "useful," "original," and "quality" the dominating part of your visitors (say, 70-80%)? Make most out of your remarkable text writings peppered with enticing and high-quality multimedia data – and the searchers visiting your website or blog will be unlikely to "bounce", closing your tab to keep browsing till they find exactly what they are looking for, besides being offered in a more appealing way or better/handier visual look.