How do I quickly get my website highly ranked with SEO?

Before you get into SEO, you need to make sure your website is properly developed and has a good infrastructure.
You can start off by implementing meta keywords and make sure your website is up to date and viewable by all users on different systems.
There is no "one way" to rank it..
it requires constant practice and more than one way to rank it..
and it's very natural to take some time as well! Hope this helps!

There is no magic stick to do this Always, start with good plan for your website Support your site plan with high quality SEO strategy, start with Technical requirements ( Friendly design with lightweight pages) even your host provider is apart of this.
High-quality content is your secret to do all your next steps, High-quality content means all the text on your site (Articles - Image alt tags - Meta tags - Descriptions ...
) after buliding your high quality site, it's the time for building links and submitting your site to Search Engines

the best thing you can do to get high ranking in Google optimize your landing page with the keywords and page title with related content all of these together will give you high quality score in Google because Google calculate your quality score for each keywords every time when the search query matched with your keywords

Well your question has hidden answer i.e.
"How do I quickly get my website highly ranked with SEO?" You stated you have a) Domain Name b) Your Website and now you want your website highly ranked with SEO.
Answer is: Just buy a Domain name with your Targeted keyword.
Google will match the searched keyword against the Domain name and the chances for coming up are high and easy and quick.
For example: You are selling "Cheap Cars in Brisbane" now you should buy a domain name CheapCarsinBrsibane.com.au and write 500 article content on that page by using the same keyword 2-3 times (obviously use some secondary keywords too).
and you will rank highly on search engines easily with so less work :) because half your task is done by a DOMAIN NAME you own.

"SEO is only not seen as rocket science BY THOSE WHO ALREADY KNOW IT” ~Danny Sullivan Before putting some short tips/steps, lets state and assure that SEO is an on-going process that never ends.
Here is a short tips to start from:
1- Keywords Research: Keywords are at the heart of SEO, and selecting the right ones can make or break your SEO strategy.
Compile a list of about ten keywords associated with your product or services.
Plug these keywords into Google’s Keyword Tool, and find variations that make sense for your business.

2- On-page SEO: Include these keywords in your: URL, Page Title, Meta tag description, On-page Headings, throughout the page content, images name and alt.

3- Off-page SEO (Link Building): Meaning to have links in other websites linking to your website.
Can be done through: Co-Marketing, Social Media, PR, Directories, and the most important: having interesting content that other website would love to link to.

4- Use Google Webmmasters tools, to tell Google you're there, and add sitemaps.

5- Monitor, Analyze, and Improve.

I don't think there is a way to quickly get the high rank, it's a lot of work and patience.
Google has great content and guidelines to start with SEO, it's really useful, and it's important to focus on every word they are saying, because they own this, and they know exactly what they are talking about, please don't miss any word, link or video.
It's good to start by subscribing to Google Webmaster Tools, and adding your site there, they will guide you step by step to make your site works well in search results.
Here is the link to Webmaster Tools: http://support.google.com/webmasters/ Then go to Web Master Academy, it's a great tool from Google, it has a checklist from 27 items, complete them all, be patient, give your changes some time, then you should be more satisfied with your site rank.
For the quick solution, I think submitting a sitemap on Webmaster Tools will boost the discovery of your pages, so this might help with going quickly.
Now all of the above are mostly about guidelines, no guarantee on the results, I think the main focus if you always want to be friendly with Google algorithms is to focus on your business and deliver good content and good value to the web.

Before answering the Question, let's comment on the question it self, How do I quickly get my website highly ranked with SEO? * No one can grantee a quickly website ranking, or a high ranking, * To get the best ranking desired, you have to be updated with Google guidelines, Panda Updates and Penguin updates, also any other allowable tricks, * There are 2 things you have to care about while doing SEO, On-page & Off-page * Keep away of Black hat SEO * Content is the king, keep posting new and fresh articles, where Google is in love with new content.
* Create a smart Back linking strategy.
* I think you should try techniques and be patient.
* No one knows when Google about to update the ranking.
* Never ignore Social Media impacts