These data sources should be used together. Use Google Analytics to gain an immediate understanding of how busy your deployment is, sources of traffic and indication of engagement (bounce rate, time on site, page views per visit).

Want to analyse the people that "bounce" (those who do not extend their visit beyond the initial landing page). Create a segment in Google Analytics for bounced visitors only. Where are they coming from (source)? What landing page are they bouncing off? Could it be a mismatch of what the visitor was expecting and what they were shown? Look at bounce rate by country - perhaps those places with weaker internet connection are bouncing more because they simply cannot load the page in a reasonable amount of time. 

As for the social media data points as Heather mentions above, these should be analysed in conjunction with Google Analytics. From my own experience, I bet that if you looks at your deployments traffic sources, visits from Social media are likely the most "engaged" - they bounce less, spend more time on the site and perhaps look at more pages in each visit. The insight you can take from this is examining which posts are producing the most interest and most engaged visitors.

Looking beyond engagement you may want to consider what the goals of your deployment are. Is it to crate awareness (perhaps simple unique visitors is a measure of success in that regard). Perhaps your deployments goal is to encourage people to take action of some sort. In that case report submissions may be a good metric. Is it to generate users or email addresses? Perhaps then email sign ups via get alerts is your measure of success. Similar, the goal could be prompting people to contact you - perhaps "contact us" form submissions then are your key metric.

The chances are that a combination of all of the above will offer insight on how successful your deployment is. The key before doing any analysis what so ever is to ask yourself "What is the purpose of this deployment". From there decide on your metrics (could be visits or social media mentions for awareness or other key actions for goals).

Once you have defined your indicators use Google Analytics, Facebook insights or other data sources to measure your deployments performance every so often, perhaps monthly.

Tip. Try to keep your number of metrics low. It's easy to go down the rabbit hole and start trying to measure everything. Ideally 4 or 5 metrics should give you the "pulse" of your deployments performance. Anything more and it's easy to lose sight of things.