Yes, you'll definitely use muslins loads later.
Newborns can poo loads - something like 6-8 times a day, and although it's liquid it's quite sticky. The muslins don't really need to absorb poo though, just hold it in place, and they should cope with absorbing wee from a newborn.
Liner-wise, I used fleece liner until DS was weaned. Then I wouldn't deal with the solid poos (partly due to morning sickness) so I swapped to paper. But really with fleece liners you do the same as you would with a muslin directly. Sometimes you can tip it down the loo, sometimes it's too sticky, so then I get the pooey bit under the flush and flush the loo, and that normally gets rid of the poo. You can get very nice, soft paper liners if you fancy them though.
My washing machine is pretty rubbish, but it copes. I used to set it to do an extra rinse, and once DS was peeing lots sometimes a prewash too - they only add about an extra 15 mins to my 50 min cycle. I am more regular in running an occasional 90° wash though, just to give it a good clear out.
The good thing about muslins as well is that they're fairly cheap, so if you ever have a total poo explosion and you can't get it off or face putting it in the machine you can just bin it. I would look on ebay as sometimes people sell lots of muslins in a bundle.
That said, DS stopped pooing loads quite early, and for about 3 months only pooed once a week/every 10 days, so it was quite easy to deal with his nappies.
Also, meant to say, if you go with eco bumbles I would also get a couple of cheap towels from ikea or matalan (or use old ones) and cut those up into booster sized pieces - that way you get terry boosters for much less money. And you can use folded muslins for boosting too, especially at a young age if you just need a bit extra.