This is a great little free tool you can use on your website or blog, StatCounter.com. (No, I'm not affiliated in any way so I'm not trying to advertise here.) I just started using it on my website not too long ago and so far it's been pretty interesting. It's not just a counter. It gives you all sorts of info on where visitors are from, referring sites, search engine keywords they used, where you ranked in that keyword search, if they found you through an image search, etc. This is helping me optimize my site by using more of the popular keywords. I'm finding out, for instance, that a lot of people land on my blog searching for hyena art and brown pelicans, which I find odd. I would not have thought of those as popular search subjects.
I think it's worth the time to watch the Keyword Analysis and Recent Keyword Activity on your site. Pay attention to what search terms seem to come up frequently and use more of those keywords in your blog posts.
One thing that seems to make a difference is naming your image files with the keywords. Don't name your barn painting with something generic like 'image01.jpg', for example. Be sure to name it what it is, plus some extra keywords, even if the name is long. Name it something like 'landscape-farm-red-barn-oil-painting.jpg'. I've gotten more visits through Google image searches since I change my file names. Even though it was a big pain to change all my file names and the corresponding HTML, I think it's worth the effort.
Other interesting information you can gain is finding out where people are coming from. If you post in several forums, for example, and include your website or blog in your signature, you can see if visitors are coming from those forum posts. If you notice more visitors coming from certain forums, and fewer from other forums, that can help a lot. It tells you which forums to spend your time and energy posting in, and which to not waste your time on.
There's tons more info you can see from this site, like which of your site pages are most popular, what images they've downloaded from your site, what countries they're from, even what browsers they're using. Lots of info! Worth checking out.