An agency made an agreement with client about SEO services, and they should start work. The best way is to have some procedure, written process steps, so you follow it and know what should be done and when. If you don’t have that, work could get chaotic and you might miss something.
Initial accounts access
First you will have to ask your client do they have Google Analytics, Search Console and Google My Business accounts. If they have, they should add your agency email as admins so you can use that 3 very important SEO tools. Also you will need Cpanel or FTP and CMS access (unless you are doing just consulting work, and client have their own coders who will implement the changes on website).
When you have all that access data, you are doing initial analyse. You analyse visits, situation on Search Console (positions, does Google bot have some issue while visiting website, situation about indexed pages, etc). You should check duplicated content also with some external tool. Regarding duplicated content, checking should be done on pages on website but also external (do the website have some content that is exactly the same as content on some other website). Then analyse links to website. Amount, relevancy (topical and geo), how strong links are, frequency etc.
The good practice also is to analyse competition, web pages that are showing on Google first page for targeting queries.
Making the SEO strategy
After full analyse, SEO agency will make SEO strategy. Don’t mix strategy with tactics here. Part of strategy could be making good links from news website. And how you will get that links is tactics. So, you will see opportunities where website’s SEO can be improved and strategically make plan about what you will be focused on while doing SEO.
By on-site changes I mean on-page changes as meta data, content, images optimisation, … but also things you will do on server side but not on actual page. That include robots.txt, sitemap, htaccess file, SSL certificate, …
Ongoing SEO process
And finally the part which will be repeated. Off-site SEO plus constant analyse to track changes and to see if some issue will appear again. You will think about providing new links here, brand mentioning, adding new content, fixing title and description tags if necessary, … This part will depends on your strategy, of course.