You can submit your site to search engines using their site submission forms - this will get you into their system. When you actually get crawled after that is impossible to say - from experience it's usually about a week or so for an initial crawl (homepage, couple of other pages 1-link deep from there). You can increase how many of your pages get crawled and indexed using clear semantic link structure and submitting a sitemap - these allow you to list all of your pages, and weight them relative to one another, which helps the search engines understand how important you view each part of site relative to the others.
If your site is linked from other crawled websites, then your site will also be crawled, starting with the page linked, and eventually spreading to the rest of your site. This can take a long time, and depends on the crawl frequency of the linking sites, so the url submission is the quickest way to let google know about you!
One tool I can't recommend highly enough is the Google Webmaster Tool. It allows you to see how often you've been crawled, any errors the googlebot has stumbled across (broken links, etc) and has a host of other useful tools in there.