Given you have a good idea of where to start and are fairly confident you are speaking the same language as your client, jump start research by generating related keyphrases and long tail variants with the ever so easy to use Google Autocomplete. This tool makes predictions based on what you are typing that are a reflection of Google search activity.
You can improve your site speed by a ton of methods, but the overall goal should be to test your site from different geo-locations using a tool like pingdom and then attend to issues. You could go for a CDN provider like Cloudflare or install caching plugins that speed up your site by reducing database queries and therefore the server load. Choosing the right hosting company for you is a critical decision and is based on many factors including your CMS, expected site traffic, and what your goals are for the site amongst others.
After diagnosing your site through the different facets of the search engine (Google), it’s time for you to check your website as an entity. The tool we’ve always used to check on our site’s onsite status is Screaming Frog. We’ve always used it as the websites we handle grow larger as the months pass by. You set the parameters and it’s even capable of crawling/compiling outbound links to let you know if you have broken links. Here’s what the overview looks like:
KWFinder was developed and created by Peter Hrbacik. He is amazing at providing great support for the tool. They have live chat on their website, which I have used quite a few times during the day. Also, their email support is also awesome. Below are a couple email conversations I have had with Peter. In this first email I suggested that they make the category headers clickable. Peter responded within 24 hours and said they will probably change it. And a couple days later, the change was implemented.
Of all the tools listed in this article, Moz Link explorer is an old one & quite popular. If you want to compare backlinks between two or more domains, Open Site Explorer is worth trying. This tool works best when you have a paid account of SEOMOZ though a free version of this tool is good enough to get you started checking the backlinks of your site and the sites of your competitors.