Download the raw log file from your server via CP (if you have access to it) and just do a simple "find" for "googlebot". Likewise for all other bots. You'll see when it came, what it spidered, when it left, etc.
Search for your URL in Google and see if you can see if your homepage is indexed. Alternatively, do a site command. ie: site:http://www.domain.com/
yep dcristo is right, to make it simple.. you just write your whole URL. If your site was in the results, then congrats and if not just wait till Gbot shows in your log