confession, I'm a 20-something who loves sewing (hand or machine). And knitting, cross-stitch, and quilting. I am also insanely busy with school, work, and applying. You make time if you like something.
Do I sew a whole lot during the school year? nah, but it is fun. It's nice to get something concrete that you made from scratch. I'd recommend starting with hand sewing, naturally, since you don't have to go out and get a machine or anything fancy. I started when I was little, just making doll clothes, but to bring that up to an age-appropriate level, start out learning the basic running stitch and the blanket stitch (google, or libraries should have books) and practice by making something really simple, like a bag or a pillowcase or something. That way you won't be too bummed if it looks like crap. Even if it does look like crap, working with your hands to do tiny things will get you used to it. Really, just pick up some fabric and see what you can do with it. No real instructions or patterns or anything necessary.