Do the home page first, because your internal links will pass some juice down the line. As far as the other pages, you'll find some that will need an extra push. As far as your competitor ranking 16,000 pages at #1 - I would like to know what kind of keywords they are targeting. I mean, you can rank #1 for "chocolate pizza" pretty easy, but are you going to get any traffic?

Off page SEO on each of those pages certainly wouldn't hurt, but might not really be needed for all of them because some pages might not really need as much coming back to it as others. One thing to remember is that juice will drip from one page to another with a simple dofollow link. It doesn't matter whether that link is internal or external. With that in mind, I'll use web hosting as an example on how I would think about doing it...

-Homepage would link to highly competitive sub-pages like "shared hosting" "dedicated servers" and "vps hosting"
-Each of those pages would contain a few in-content text links to somewhat easier sub-pages like "wordpress hosting" "joomla hosting" "linux vps" "windows vps" and "linux dedicated servers"
-Then, each of those would have links going to somewhat easier terms like "cpanel hosting" "cheap linux vps" "affordable linux vps"
-Keep the process going..

Once that kind of structure is setup you can begin doing offpage SEO to your homepage or first set of inner-pages. From there juice will start flowing to your other inner-pages because your linking to them internally.

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What bdenzer said. Concentrate on the homepage first, your inner pages are linked to it anyway, so that should help.
Also, you don't need to focus on all 20,000 pages. You could focus on the ones you think are more important or the ones you think are easier to work on.

You don't have to do off-page SEO for all your separate pages. If you focus your links towards your home page, that in turn will pass the "link juice" down through your other 20,000 pages as long as you have good site structure.

Although with that said, you can focus links directly towards an internal page it it will do better than the others in the SERPS (depending on competition for the keywords on that page, obviously). You can use this to generate more traffic and profit depending on the individual page.

For example, if I ever have a page that is converting better than others on my Amazon Affiliate site, I will direct links for a couple of weeks to that one page to give it a higher position in the SERPS so I can make the most of the higher conversion rate.

First of all what kind of content have 20,000 pages. Is images, videos?! If it's crappy scrapped content it doesn't even worth answering the question.

IF the website have some quality on it focus on offsite SEO for homepage and make the website more user friendly and easier to share.
Any gallery kind of website is based on making people share their content. Otherwise that is a useless number.

How on earth you could make legit offsite SEO for 20,000 pages.

Agree with the comments above on the home page being the focus and that the internal pages gain from that. Since you've got so many pages, what would be the next tier down of pages that get the most traffic? Perhaps break your site into subsets and only focus on the highest potential traffic for now?

Makes such a mammoth job more manageable.

Thanks ! a ton everyone here.
I have also started the sequel s to this thread (Confusing SEO - part 2, 3), hope you all would love to contribute about my site.

Link Mentioned.