Both mooshrooms and blizzes/blazes can "easily" be gotten if you look into what some items do.Something which sprung to mind with me last night when I was doing a bit of playing, is that luck with loot bags can have a big impact on your play. I don't mean things like getting your hands on a full set of armour or anything like that, I'm talking about the stuff that you can't craft. Spawn eggs and Hives are an immediate example (yes, I know that to a degree you can craft hives).
I got the spawn eggs loot reward, which had two mooshroom eggs, so I could spawn them, breed them, and then of course shear two of them and get cows. I didn't have to worry about trying to get a grass area to spawn them (which I've had very limited success with at the best of times). The mooshrooms also went a step further as they can be milked for mushroom soup which made that go with the flow quest really easy.
I also got a blizz egg in that same bag, which means that I can spawn it, trap it in a re-usable safari net and then chuck it in an auto-spawner to get blizz rods out of them. Again, this is going to make the gelid cyrorothium (sp?) quest a lot easier.
I'm a bit torn on these sort of rewards, as I know that it's luck, but I also know that you end up with getting a lot of loot bags, so there is a reasonable chance of having some quests easier because of your rewards.
Only open spoiler if you want to know how since its quite literally a spoiler.
Mooshrooms can be gotten as soon as you can sift larger amount of sand and dirt by getting grass seeds for a passive mob spawning area(non desert/ocean/river) for a cow.
Followed by sifting for ancient spores which when used on cows makes a mooshroom(mycelium can be gained through other means as well).
Blizzes can be found in the winter biomes quite easily by simply having snow and ice blocks(not sure of this but empirically this seems to be the case) present they will spawn regardless of light level, but following other normal hostile mob spawn rules.
Blazes can also be gotten from the obvious location of the nether fortresses as per usual minecraft.
All of them can be captured with golden lasso(non hostile)/safari net for storage to reduce entity cap or to put in a minefactory spawner.
Followed by sifting for ancient spores which when used on cows makes a mooshroom(mycelium can be gained through other means as well).
Blizzes can be found in the winter biomes quite easily by simply having snow and ice blocks(not sure of this but empirically this seems to be the case) present they will spawn regardless of light level, but following other normal hostile mob spawn rules.
Blazes can also be gotten from the obvious location of the nether fortresses as per usual minecraft.
All of them can be captured with golden lasso(non hostile)/safari net for storage to reduce entity cap or to put in a minefactory spawner.