What you measure depends on your objectives at the start e.g.
If you’re using social media to boost your Google ranking, use Google search. Search Google for ‘Google’ then click ‘Everything’ and ‘More’ to search just blogs or ‘realtime’ tweets. (If you ‘vanity search’ your website, name or keywords, be sure to click your link/s and spend time browsing around your site so Google treats it as a ‘real’ search.)
If you’re using social media to build more inbound links to your site (a good way to boost your Google ranking) you can monitor success with backlink tools such as BackLinkWatch.
To analyse your web traffic, use Google (or Yahoo) Analytics. You get numbers, graphs and charts at an unbelievable level of detail, and it doesn’t even cost anything.
If you want to grow your network, simply count your likes/fans on Facebook, followers on Twitter, contacts on LinkedIn, and subscribers on your blog.