It’ll take some time studying Flarum’s styles. But, my tip for you is think minimal. For example, for a tag background or button style, to get the same look as Flarum, it’s pretty much as simple as adding a muted background color, a little padding, and then border radius for the corners.
It’s not exactly like Flarum, but I’ve used this water.css in the past as a start. This water.css instantly gives any website basic, minimal, and good looking styles.
One thing you can do to help match the look is the fact that Flarum uses Font Awesome for icons. So be sure to use Font Awesome in the other part of your site (if that part of the site needs icons).
Edit: Actually, the more I look at water.css again, the more I think it is very close to Flarum’s look. So, you might want to try it. Maybe tweak the water.css a little to get it even closer to Flarum as needed.