The increase in visitors doesn't increase rankings... Increase in rankings increases the number of visitors...

Thanks. However, I want to know some baseline numbers to see how long it typically takes to improve rankings. Got any wisdom to share here?

Guess this answers the question somewhat:

How quickly will I see results?

If you target long tail keywords you can see results pretty quickly but always remember SEO is a long term strategy not a set and forget thing.

If you’re after more competitive keywords prepare to commit to it for at least three months of consistent effort.

Your question doesn't have a specific answer. However, from my experience, I see results quite fast (within 1 week, on page 2, some manage to page 1). As other people said, it depends on your keywords, url... Your SEO method matters too. Footer links work great, so do private blog networks/web 2.0. The important thing is you need to have enough unique, readable content on your links to pass the penguin, panda test

What do you mean "penguin, panda test"? Does it also depend on which page or which entry I want my search rankings to be?

I mean after the panda/pengui updates, you need to build more quality links. Sure there are a lot of factors affect your ranking but here are the ones I see matter most:
- You on page SEO
- Your backlink quality/diversity
- Backlinks to your backlinks

If you after big keyword, make sure your domain has some kind of authority too. It will be tough to rank a brand new domain for a big keyword (huge search, high cpc). Most important thing when you do SEO is patience. Many people quit when they don't see the result instantly. If you want to see the results quick, start with easy keywords first. With the right SEO, you will see the result quite quick.