Wow. This is too big a topic to cover completely, so I will try and focus on your question with something of a mini-tutorial.
When you do a site:mydomain.com in Google, you will see your site listed somewhat in order of importance according to what Google has found. Often your sites home page is listed first, but not always. If it is not, do not worry about it- it is not an indication of a problem. When you search using a keyword, then you run the risk changing the order to try and satisfy the search query. In this case, it is possible that your sites home page will not show up at all.
Much of what is returned relies completely on how your site is organized. For example, if you run a blog where your most recent and popular posts have links and snippets on the home page, then that page will likely perform differently periodically as the links change. If you have a fairly standard website, then it is likely that the home page will not have much importance in the search engine result page (SERP). And that is okay. Use it as a bit of a landing page and change it from time to time. As well, make sure that your most important content is linked from your home page and that your navigation will help users find the content they are interested in.
As far as ranking in the classic sense, that is to say PageRank as Google has defined it, each page performs according to the topic popularity, how the page was organized. how the page was optimized, internal linking, any inbound backlinks, and most importantly, the ever changing search terms that are %15 unique each day, that is, never searched before.
If you have a traditional site, you need to organize it into topic and sub-topics and research the topic keywords that apply to each page. This is a significant planning effort that must be done periodically since search terms are constantly changing. However, there are the long-tail keywords which have performed consistently over a longer period of time. Page organization, optimization, internal linking, inbound backlinks, and just plain dumb-luck sometimes yield positive results. Just make sure you are optimized for not only actual search history, but what your audience expects. Keyword tools can sometimes steer you in the wrong direction if your topic is not main stream so be vigilant in your research.
Now here is a secret. Content is king has within it a duality. Quality content is required as well as compelling content. But here is where is gets complicated. Quality and compelling content is not enough. Diversity within a singular focused topic is also important. If you can write 30 pages on a topic, then you must write 300. This is why blogs are so successful. If you notice that most blog posts are short simple single thoughts that follow a theme each and every day. Most posts hardly say anything at all. It is the topic keyword diversity that is the goal. I hate blogs in general because most are junk; I prefer deeper content. But I think you get my point. The more keyword focused diversity you can create without duplication of content, the better. (Sigh) But please create deeper content as much as possible. Deeper content lasts longer.
The last and most important ingredient is time. Lot's of time. Search engines are notoriously slow and so are users. When creating a new site, expect about a year before really gaining an audience. Part of that process is allowing time for people to find your content, share it, and link to it. The more this happens, the more people find your magnificent stuff! And if your content is magnificent, and I am sure it is - you are here aren't you(?) - then you are a pretty smart guy, it will gain traction slowly at first, then take off. Eventually you will see how pages perform and you will get it. You will begin to really know how to make your pages perform optimally. And remember, it is pages that perform- keep focused on making each page perform as well as you can.
To go back to the home page. Using Mister Peabody's Wayback Machine for a minute, the term PageRank has changed from the first days of Google. While Google always measured links from page to page and ranked individual pages as a result, the term PageRank, from the public's point of view, originally applied only to the site. This was supported by the Google toolbar. Before anyone tells me how wrong I am, read on. In the early days, PageRank was use as a synonym for SiteRank. This was unfortunate because the confusion remains. We talk about PageRank for a site all the time, when in reality, PageRank remains on a page by page basis. However, a SiteRank still exists. As individual PageRank is earned and indeed as a site ages and gains trust, the collective PageRank will help the home page perform better. It is the competition of the site and not the pages so much, that result in increased performance over other sites. Like in Genesis, individual page performance begets site performance begets home page performance.