There a more easy and reliable way I can find artists taking commissions or do I just search randomly on the sites, see their furaffinity, and find out?

You can make a post on forums (FAF, here, DeviantArt, etc) looking for artists to commission. You can also search for "commission" on FA utnil you find some pictures you like and check out the original artists to see if they're open for more commissions.

If you post a thread inquiring about commissions, be sure to list your budget, what you're wanting to be drawn, and preferences such as colors/styles.

Well for start, find an artist that you like (how he draws) then check his pages like FA...etc

picarto.tv has a bunch of artists with open commissions, there's a place to start. after that, follow the artists you found/liked/commissioned on twitter, then check out the art they retweet and check of those artists take commissions, too. soon, you won't have any trouble finding an artist to commisssion.