
The main thing about taking care of pokémon with opposite typings is that it’s important to keep in mind that as a trainer, you’re very likely going to be doing this anyway. Unless you’re only raising one pokémon or you’re attempting to become a type specialist (and in some cases, not even then), you will likely catch and raise a wide variety of pokémon at once in order to build an effective team. Otherwise, you may struggle to be successful in any given pokémon league, as you will absolutely encounter trainers, particularly gym leaders, who will carry a vast assortment of pokémon and are armed with enough experience and strategic skills to make limiting yourself to a single type a challenge.
And even if you aren’t a trainer, it’s very likely that if you own more than one pokémon, their needs will be completely different, especially if their types aren’t the same. This is true, even if their pokémon types aren’t actually incompatible. Raising a meowth is nothing like raising a magikarp, and it’s a good idea to keep this in mind as well.
With that said, the trick to raising multiple pokémon with different types is to not think about the way their types interact with each other. Rather, think about pokémon as individuals. This will help reduce the confusion or intimidation you may face when doing research into what your pokémon need. That is to say, even if raising pokémon with different types may seem like an arduous task, if you split it up into the needs of one pokémon and then the needs of the other, it becomes quite manageable.
As you do your research, make lists of what you would need, from space requirements to basics for each of your pokémon. Cross reference these lists to see if you can double up. For example, Alolan vulpix and standard vulpix actually have the exact same diet, so you don’t need to buy special food for them separately; this will help reduce the cost of feeding them and the amount of space you would need to store their food. For another example, most ground-types need a sandbox or other structure set up to allow them to give themselves dust baths for routine grooming. Coincidentally, bird pokémon also need dust baths for parasite removal, especially if you primarily keep them outside (as you would with larger birds, such as braviary or pidgeot). Keeping one dust bath for both your sandslash and your pidgeot will keep them both happy.
Of course, the other most important thing you need to know is work with your breeder or shelter. Some pokémon don’t fare well with members of a different type. This has nothing to do with their species and everything to do with the individual (and usually how they were raised previously to coming into your care). Even if Alolan vulpix and standard vulpix actually get along quite well normally, you could come into trainership of a specific Alolan vulpix who is terrified of fire-types. It’s still possible to train that Alolan vulpix to not fear fire-types, but this will take a fair amount of time and effort and certainly changes your requirements either way. (For one thing, you will quite literally need to create a space that allows your Alolan vulpix to live in your home entirely separately from your standard vulpix without stressing either of them out. That’s a bit of a challenge for most trainers.) So always ask questions to gain an understanding of your pokémon before you adopt them.
Other than that, it’s all as I’ve said a matter of doing research into the specific species you wish to adopt and train, but for Alolan vulpix and standard vulpix in particular, this post may be a good place to start: http://bills-pokedex.tumblr.com/post/154749948831/what-are-different-care-methods-for-alolan-vulpix
Best of luck!