chicken eggs take 21 days, mallard-derived ducks & turkeys 28, muscovies 32 (or 35?), geese 28-32(? or maybe 35?). Ducks require higher humidity than chickens, geese require even higher humidity than ducks plus a cool-down/misting every day.
Personally, I would start with chicken eggs (they're pretty easy and take the least amount of time). Then move on to duck (they're pretty easy too, but that extra week can feel like an extra month) or turkey. I've hatched chickens, ducks and turkeys and I'd love to try geese this spring, but they seem like really high maintenance eggs to me; they're a little intimidating.
It is possible to hatch some eggs together, although it can be a little tricky.