Ask for a technical and search audit for your site to learn what they think needs to be done, why, and what the expected outcome should be. You'll probably have to pay for this. You will probably have to give them read-only access to your site on Search Console. (At this stage, don't grant them write access.) Your prospective SEO should be able to give you realistic estimates of improvement, and an estimate of the work involved. If they guarantee you that their changes will give you first place in search results, find someone else.
Many search engine marketers think who you link out to (and who links to you) helps determine a topical community of sites in any field or a hub of authority. Quite simply, you want to be in that hub, at the centre if possible (however unlikely), but at least in it. I like to think of this one as a good thing to remember in the future as search engines get even better at determining topical relevancy of pages, but I have never actually seen any granular ranking benefit (for the page in question) from linking out.
The ranking of your website is partly decided by on-page factors. On-page SEO factors are all those things you can influence from within your actual website. These factors include technical aspects (e.g. the quality of your code and site speed) and content-related aspects, like the structure of your website or the quality of the copy on your website. These are all crucial on-page SEO factors.