Apart from the usual google Analytics and Google Webmaster tools this article will give you some insights into what tools can help you figure out and optimize your search engine marketing strategies. Their are no surprises for me in the list but its a good list and the descriptive summaries give a quick idea of why the item is of use.
Google speed insights is on the top of the list and its one of the favourites. It picky and if you attempt to implement all of its suggestion you can easily get overwhelmed. But if you have time do it and it will make a huge difference not only on the speed of the site but also on your competitive ranking.
Check out the list at https://blog.bufferapp.com/free-seo-tools
And buffer is a good tool. I use it to help distribute these post to my social network. The basic system allows 5 sites. You can schedule post out and determine how many a day and at what times. Their are other tools that give you better integration and take less time but it a good start.