I would recommend having these three tools and you will be able to do a lot.

1. Screaming Frog to Crawl your site. Alternatively, Sitebeam site crawler works for free.
2. SEMRush or Ahrefs to look at backlinks. SEMRush is cheaper so I will recommend it, although I prefer Ahrefs. These tools also provide audits and rank tracking.
3.If you are a local business, check out Yext or Brightlocal for citations and NAP consistency.
4. If you are heavily focused on content, you could benefit from Content King or Buzzsumo.

If I had to choose the most barebones, I would say: SEMRush/Ahrefs + Screaming Frog. Master one tool instead of have a lot of others.