One thing about strumming is that it can become a dead-end. It doesn't have to, but I think it often does. If you're going to strum, one thing you might want to practice is hitting only some of the strings. I mean particular ones, like the top four or 5, 4, and 3. I remember finding this difficult, but now it's no problem for me at all. I usually pull multiple strings rather than strum, but you can't do this with a flatpick. The face of your guitar will thank you for the improved pick control.

Even if you only want to accompany, I think it would be a good idea to practice playing the melody, especially by ear. It seems like playing melodies on the guitar should be easy, but it isn't. At least, I still have trouble playing them accurately, i.e., without making mistakes, even though I practice a lot. I should have started learning to do this sooner.