Yeah after a while, looking after your animals and stuff does get easier. I recently started a new game after leaving my old file so long I forgot everything and have just started Summer... I'm struggling atm to water all my crops and take care of my two cows as I only have the copper watering can xD I miss my golden and ultra cans...
(I never actually got the speedy milker last time though. Exactly how is that one faster? As the milker is pretty fast already imo)
And yeah, you can have three barns and three coops in the end... but tbh, you REALLY won't need that many animals (and if you did, you'd have no time to do anything at all, even with the speedy tools!)
I'd fill up one chicken coop with maybe 2-3 regular chickens and the rest Silkies, and leave it at that (even then you'd be rolling in 50 eggs a day at maximum productivity!). For animals... really, you only need one of each (two maximum if you want one to produce Great products). Once you've maxed productivity, each animal will produce 5 items a day... and do you really need four cows, five sheep, three yaks etc all producing 5 items a day? Nope. Just one animal will be more than enough. Considering there's 8 different types of big farm animals, if you have just one of each, you only need one barn and it'll make your farm life so much easier. Especially once you unlock the east side of your farm - that's got a raised area for lots of crops, and you can water them all at once just by moving a rock at the top. Once I have that area I move most of my crops there and keep the main part of the farm for stuff that doesn't need watering, like trees and mushrooms and bees.