Is there a direct correlation that a high PR will translate to my URL being shown on the 1st page of a search? Can someone tell me approx. what search page that a PR3, PR4, PR5, etc will appear on?
It doesn't have a direct involvement. No they can't, it depends on the term, on-page factors, anchor text to the page etc rather then PR.
Can't say I've ever seen a hard and fast correlation but in general there does seem to be PR groupings based on competitiveness of the search term. PR0-PR2, PR3-PR4 then PR5+. The more competitive a term the higher your PR must be to get on the first page of SERPS. Try a highly competitive term like "credit cards" (92 million serps)and then check the page rank of sites on the first page. Then check a less competitive keyword in the same category like "prepaid credit card" (2 million serps)and check the pagerank and you will see what I mean.
If that were true, for the Credit Card term #1 would have the highest PR, #2 slightly lower #3 slightly lower and so on. But if you have a look it isn't the case the PR for the domains is scattered. Google the term "search engine" and you will see Google's PR10 being beaten by a PR7. When you start doing a more advanced analysis on things like keyword density, allinanchor, allinurl, on page elements like Title, Metas, <h> and <b> tags, where the keyphrase appears in the document etc you start to see the "real" reason they rank rather then what it appears on face value of a green mark on the toolbar.