It depends (as everyone has said) what direction you take it in. I was a 4-H sewing member for 9 years, then worked at fabric stores while in college/grad school during my breaks - I probably sew 50% of my cotton shirts, all my skirts/dresses, and probably about 50% of my kids' clothes. And many of the gifts I give for new babies/holidays somehow end up involving sewing. I also quilt (and embroider, but not with a machine).
I sew big receiving blankets as my go-to baby gift. If I find the flannel when it's on sale 50% off, then they're not too expensive. I stocked up for YEARS on baby flannels so I could do that ... in the end it's probably still a pretty expensive gift, but people ask me for them, too, so I know they're appreciated. Storebought ones are cheaper, but they also don't work for big babies, or for any baby past about 2-3 weeks of age; I've friends who swaddled their babies 'til they were a year old (!) in my receiving blankets. I also give aprons as gifts, or will sew cute things for nieces/nephews, etc. Another gift I've given often is flannel pants (plaid flannel or print, for lounging at home, comfy pants) - often by request of siblings etc.
My kids are skinny - and until they hit 4-5 years of age, nearly impossible to find pants that would fit in the waist. So I made LOTS of elastic-waist pants (flannel, corduroy, denim) for them for those ages - and lots of dresses too. That was less "saving money" than simply being able to clothe them! I also modified some store-bought jeans for them as others have described above.... And modified other patterns to make fitted jeans or etc. for them (size 2 waist, size 4 length...)
I recycle clothes. I've got a big stack right now, in fact, of t-shirts I can't wear anymore (now that the waists on jeans are so much lower - perfectly good t-shirts which are simply too short, so no good if I were to donate them anyway) - which have been becoming shirts for the girls to wear. Simple, easy, cost me nothing but thread. I plan to do the same with all my huge old wool sweaters sometime this winter/spring.
I made my own birdseye flat diapers when dd1 was born, and made my own cloth wipes too (and gift those to friends who are cloth diapering). The wipes are simple, just use scraps from the receiving blankets! Or from flannel pjs or etc.
I make the kids' pjs too so we don't have flame retardant pjs. I use cotton flannel, knit, or fleece - no chemicals. Nice. Also - found some great washable satins etc. on clearance and intend to make a bunch of play clothes out of them.
I sewed my own wedding gown, one of my sister's wedding gowns, and have altered several other wedding gowns and bridesmaid/formal gowns over the years. Once you become a competent, capable sewist you can move from sewing for yourself/your family - to sewing for others. When I worked at a fabric store, one of our clients just did zippers. She would replace zippers for people. She had it down to a science, and made a good profit doing it, because zippers (especially in jeans/khakis etc.) are typically not something people really enjoy doing, and not something many people feel comfortable repairing themselves, "But it's a perfectly good pair of slacks!" Others have already mentioned etsy etc. I don't think anticipation of making a living with couture sewing on etsy or local mending repair business are a reason to begin sewing - but they could become a benefit of learning to sew.
It's really nice to have clothes that fit perfectly. Christmas dresses which I like vs. the ones I see at the store .... I agree that knowing that at least in their construction, the clothing didn't use sweatshop labor etc. is nice. We try to do as much "fair trade" as we can, and this cuts out at least the manufacturing process of the clothing etc. that I make.
That said ---- I have a huge stash of fabric, and I like high-quality fabric, and have way too much of it. I am in Year Two of trying to work though my fabric stash and make a significant dent in in. So you can spend a LOT of money on lovely fabric, and it's especially not worth much if you don't ever get it sewn up!
But, the girls and I have unique clothes and receive many compliments on them. Clothes that really fit what we like and who we are, and are attractive. And it's a very enjoyable hobby for me - my eldest is 7 and already learning to machine-piece for quilting, so it's a hobby that I'm able to pass on and enjoy with my kids.