You should see an initial improvement in rankings fairly early on, particularly if your site was not previously visible for the key search terms you have now selected.
How fast you move to the first page, however, will depend on a number of factors. Typically, the stronger the competition for a key search term, the longer it takes to improve your visibility for it. Putting more resources towards this can make a significant difference to the time it will take your site to climb in the rankings, but if your competition has been around for a long time or has benefited from a strong integrated marketing campaign, it can be especially difficult.
Though they are a good indicator of initial progress, positive ranking changes will not immediately translate into more traffic. It won’t be until your key pages have reached the first page of the search engine results that you are likely to start seeing clicks from organic search.