We have been using cloth diapers with my DD since she was about 4 weeks old (we didn't have any newborn sized cloth diapers). I got mine as hand me downs from my sister so my initial cost was $0. You don't need to put plastic covers overtop. With poop diapers we just rinse them in the laundry sink because breastfed baby poop is water soluble. Once she is on solids we will probably put a sprayer on our toilet. You can also get flushable liners that go in the diaper so when the baby poops you just pick up the liner and flush the liner and poop down the toilet. I have 24 diapers and I do one load of laundry every second day. It is winter here so I hang the diapers inside to dry (or sometimes put them in the dryer), but once it is warm I will put them outside on the line to dry. UV rays from the sun are supposed to get poop stains out, although we haven't had any staining yet. Pee diapers stay in the diaper pail until it's time to wash them, poop diapers get rinsed in the laundry sink and then soak in a pail of water with a bit of oxy-clean baby stain remover until it's time to wash them. This system works very well for us, but everyone comes up with their own system. We have pocket diapers, so each diaper has a pocket inside where you put in an absorbed insert that soaks up the liquid. We put multiple inserts in overnight so we don't have to change her at night time. When I do laundry I hang the diapers to dry overnight, then in the morning I put the inserts in them and put them in the change table so they are ready for the next two days. Takes maybe 10 min to stuff all the diapers with the inserts. It might sound like a lot of work, but once you find a system and get the hang of it it's really no problem at all.
We also use flannel wipes instead of disposable wipes. I figured if I'm already doing the diapers in the laundry I might as well do the wipes too. I make a bum spray out of tea tree oil, lavender oil, witch hazel, jojoba oil, baby soap, and water. The ingredients costed about $30, and since I only use a drop or two if each ingredient in a bottle of spray, I will never have to buy more. It smells great, cleans her bum great, and is very gentle on her skin. The spray literally takes seconds to make, so it's no big deal. When we change her we just spray her bum and then wipe her down with a flannel cloth. Bum cloths go in the laundry pail with the diapers. I spent $20 to buy 48 flannel cloths which is more than enough.
I much prefer our system instead of disposables. Obviously it's a huge cost saver, especially since I didn't buy the diapers in the first place. But we also have not had a single diaper rash, never have to use bum cream, and never have to worry about running out of diapers or wipes. We do use disposables sometimes when we go out of the house for the day. I have a 'wet bag' in the diaper bag where I can put a dirty diaper so it doesn't leak all over the diaper bag. If DD has already pooped then I will take her out in her cloth diapers. If I need to change her then I put the dirty diaper in the wet bag and throw it in the laundry pail when we get home. If she hasn't had her poop then I put her in a disposable when we go out because I don't personally want to carry around a poopy diaper all day. We have purchased one bag of disposable diapers and have only used about half the bag in three months. It's is also supposidly easier to potty train kids who wear cloth diapers because they aren't as absorbent as disposables, so the baby gets uncomfortable more quickly in a wet diaper which deters them from going in the diaper. DD is only 3 months old so I can't attest to that theory yet!
