Hello Alex,
I've lived and worked in Slovakia for 5 years and hope to be able to answer some of your questions and concerns about moving.
You mentioned that you have decided to move to Slovakia but did not mention where. Finding a place to live is much easier to do if you moving to Bratislava. This is because there is simply more options and more information for looking for places to live. If you will be looking in Bratislava, and speak some Slovak, I'd recommend checking out the Website reality.sk. In my experience they have the most comprehensive list of available properties. If you don't speak English, check out living.spectator.sk. The Slovak Spectator, an English language newspaper based in Slovakia has just created this site to assist non-Slovaks looking for places to live.
Whether to buy or rent depends on 1) If you are legal to buy property (a Slovak citizen or EU citizen) 2) How long you are staying in Slovakia. For simplicity's sake, I'd recommend renting first. You can always change your mind later.
For international schooling, there are two schools located in Karlova Ves, a district in Bratislava. A Google search should give you the information you need. I don't know of any international schools outside of the capital, but then again I've never exactly looked.
Regarding animals, I have known Americans who had brought their pets with them when moving to Slovakia. Talk to you vet before you leave. Since these legalities are regulated, he or she should have a good idea of what documents your pet needs to be allowed to travel. Also, you can call the Slovak embassy in your home country to get specific answers.
To obtain a residency permit - and also a long-term-stay permit - (they are two separate things), go to the Slovak embassy site in your home country. There they have listed the requirements you will need to fulfill and give you an idea of how much time you need to prepare. Again, if you have questions, call the embassy. That's why they are there.
I hope that helps. I have a Website called The Foreigner's Guide to Living in Slovakia that has resources and also a blog that covers some of these issues and others that you may find useful.
Good luck!
Margarete