I am using Alexa global Rank Page Tool to analyse the page rank of our website and this helps me to increase the Seo Requirements. Another tool that I am using is the Google Analytics tool and webmaster tools. These two websites will tell the website performance, how much users are using also from which countries they are using and also the backlinks available to this specific site also tells. These are some of the SEO tools that I'm using.
I agree with Google Adwords, it actually depends on budget. Im not sure if the links will be approved. If you need a keyword research try:
Google Trends, Keyword tool (.io on the end), kwfinder, and wordstream free keyword finder.
There are services that monitor the website and where people click the most that can be helpful on long terms.
I would say depending on what are you trying to achieve there are free tools out there. They require a lot of research and than again who knows since everything is changing by the minute. Its a constant game of throne 24/7.
SEO tools make your SEO perfect every time. Here are some important tools name-
1. Mozbar (it check your site DA/PA score)
2. Google Analytic (Complete web stats and search insights)
3. Google Keyword Planner
4. Google trends
5. SEO Site checkup