If you have not setup a Search Console account - you may check if your site is penalized by searching for the title of any page or post in quotes in Google and checking if the appropriate page/post shows up as the fist result. If not - then you need to start checking the severity of the penalty. This can be done by entering your domain name directly in the search and seeing what happens, or just searching for your domain brand name without the TLD or the TLD after a space separator.

2) SpyFu: I suggest to have paid account on SpyFu. I just need to find my competitors who using Adwords and review them using this tool. It will show me what ads and keywords they are using. Note that my competitor who paid for that particular keyword knows exactly that it is important for his business including recent trends. Also using SEO feature you can input any URL and find our which keywords they are ranking for.
Checking if your site is indexed properly is essential. You can do this inside Search Console. You need to make sure that all your pages are crawled and indexed and that you don't have any 404 errors or other page indexing issues - which includes the AMP (Accelerated Mobile Page) indexing issues - should you already have submitted your AMP powered site for the mobile index.
You can find broken internal links from within the Search Console. You need to attend to each warning appropriately telling Google that you have fixed it. Having excessive 404s will hurt your site if they are really 404s, because anyone could escalate the 404s by pointing randomly to pages that don't exist from external places, which is why this is not that big of a deal - but should be looked at.
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.
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