There's a hip mobility exercise that humbles me daily, and I use it with clients almost every week to build strong hips, open tight groins and activate lazy glutes: hip openers. I know, imaginative name, right?
It does what it says on the tin, though, helping to open your hips while strengthening them, which is great for improving lower-body movement and mechanics. If you want to feel freer when you move, this is the exercise to try.