Ahrefs OR MOZ or just wordpress plug-in like YOAST

Ahrefs is nice and easy with details.

You best bet would be Ahref but Semrush is also great.

When just starting out with no money, free MOZ (supplemented with free Majestic) is the best option. At that stage you can suffice with having a basic sense of the authority of pages, and the kind of backlinks they have. Later on you can make the move to Ahrefs, especially when you are tackling ranking for more difficult keywords or niches.

I think Ahrefs are very comprehensive when it comes to their link spiders - it crawls very quickly, and it gets updated very quickly as well.
Moz on the other hand, may be a little lacking in terms of its link spiders.
Yoast is different altogether, but it is definitely a must to be installed on your websites, as the basic seo plugin.
Still, if you are looking for stat tracking, then I'd go with majestic's TF, then MOZ PA/DA before anything else - Ahrefs to spy on valuable links on an expired domain.