I think you need to think hard about the line between frugal and cheap in your situation. It can be hard to discern with camping; on the one hand, you're roughing it intentionally; on the other, you're doing it for fun so being cold, wet, and sleep deprived would detract significantly from the experience, and possibly discourage you from it in the future. I see it as a vacation to minimalism; you leave all your other stuff behind and head into the wilds with just a few well-chosen items that are not necessarily cheap, but are extremely functional and reliable and durable.
So, when I finally, as an adult, moved out of the Midwest last year to somewhere with terrain and outdoorsy interest, I splurged on the REI Joule (23F down mummy bag for women). One trip sleeping in it was a revelatory experience (temps were in the 30s). I've never been so comfortable while camping, and I always had cold feet; I think I even took my socks (wool, of course) off one night! Women's bags have extra fill in the footbox; one of the reasons to get separate bags! :) I was on top of both an inflatable pad and a foam one (RidgeRest, I'm a fan; comfy and inexpensive, but bulky for backpacking, hence the inflatable). I would definitely put a good bag at the top of "worthwhile camping splurges" list; maybe not this class of bag for everyone, but pick something that fits your specifications for temp rating, weight, and volume based on your planned usage. And as others have said, lowball the temp rating; you'll probably be comfortable 10F+ above the rating and just survive below that. Reviews can also be useful in that respect...if it's rated 32F and people are reporting freezing at 45F...well...maybe not a great choice. Nothing beats an in-person test, though, which is why I bought from REI: 1 year return policy! (I actually got another bag before the Joule, took it on a trip, froze my tush off, and returned it.)
Other worthwhile (IMO) equipment:
-headlamps
-wool socks
-hydration bladder (so much better hydrated on long hikes since my bf got me to try one!)
-waterproof/breathable rain shell
-good hiking boots (I've had mine since I was 12 and they're still going strong 15 years later...BIFL).
Incidentally, I also have a thick fleece from when I was 12 showing basically no signs of its extensive wear as a jacket for some years. This (outdoorsy) stuff does NOT wear out quickly. So choose wisely :)
Other tips:
-Care for your bag. For any bag, open it up, maybe hang it up, and air it out after a trip; as long as you wear clothes inside you shouldn't need to wash it often, if at all. For a down bag, don't store it in its stuff sack! It will get used to being compressed/clumped. Store in a large storage bag or box. I think synthetics are less sensitive to this but it also can't hurt to let them expand, too.
-Wool. Socks. Always.
-Avoid cotton as much as possible. I'll wear a standard polyester/cotton hoodie to bed while car camping (because they are just the coziest thing ever and the hood keeps my hat on) but only with synthetics underneath. These don't have to be specialized, technical clothing; just check the tags.
-Wear a hat to bed
-Never wear your top layer from the evening to bed; you're likely bringing condensation (/dew) into your sleeping bag, which will keep you damp and chilly all night. Change before sleeping if possible. (I've slacked on this and definitely noticed a difference!)
-Stuff a T-shirt with clothes for a pillow. I've even used my rain jacket as stuffing. Comfortable enough for camping!
-for those with kids: the foam tiles with interlocking edges (sometimes with alphabet, etc?) can serve as tent flooring! Not cushy but insulating. Not great for backpacking.