Step 1: Assess your server's economy
The Undermine Journal (TUJ) is a wonderful resource for comparing AH prices across servers. First, look up your server: https://theunderminejournal.com/ Now, in the item/seller search bar, enter the name of the rare pet you would like to find. Once you reach the item page, scroll down to the middle of the screen where you will find a chart labeled "Other US Markets." Here you can compare your server's prices to those of every other server in your faction and region.
Do your server's prices compare favorably? Note that you can also see the average current market price and mean market price for the item for both your server and across every other server (in your faction and region) in the table at the top of the page.
If the item you are looking for is available on your server and your server's prices seem reasonable, then you just need to get some more gold

Step 2: How to deal with a terrible server economy
Unfortunately after checking TUJ you realize that your home server has terrible prices or it is unlikely you will be able to find the pet you are looking for on the AH. Luckily, you still have several options.
Option 1: Transfer gold to the opposing faction. Go back to the realm selection page on TUJ and this time select the AH for the opposing faction. Are prices and availability any better? If you are simply on the minority faction of a high pop server a simple solution may be to eat the 15% neutral AH cut and transfer some of your gold to the opposing faction.
Option 2: Create a character on a high pop realm. Certain high population realms like Illidan-Horde, Mal'Ganis-Horde and Stormrage-Alliance are known for their massive AHs and great pet prices. It may be somewhat time consuming, but in the end, leveling an alt on one of these servers can be a great way to get the best possible prices for your pets.
Option 3: Trade goods, stones or pet leveling services. Even on a terrible AH you will occasionally find relatively good deals. Consider buying duplicates of pets you already have or TCG mounts to trade for the pets that you want. Or if you do not spend much time on the AH, consider offering pet leveling services or any extra flawless battle stones you may have in exchange for new pets. There are almost always people on the Pet Trading forums looking for stones and pet leveling, and it can be a great way to trade for pets that might be vastly overpriced on your home server.
Step 3: How to make gold quickly
There are many resources out there if you are struggling to make enough gold to purchase the pets that you want, or if you don't have much time available to devote to making gold. Check out the forums at The Consortium: http://stormspire.net/forum.php or look through some of the blogs featured on the front page of your realm's section on TUJ.
As a pet collector, you also may have particular expertise on the relative value of different pets and pet breeds, which you can use to your advantage to flip pets on the AH. Consider creating bank alts on various high pop servers to exploit differences in current pet prices cross-server. If you get to know what an item should go for on a given server, you can often quickly flip underpriced pets for several thousand gold profit. Then when you see overpriced pets, try listing your duplicates and see if someone will bite. Just be careful not to start flipping pets until you're comfortable enough with what pets will sell and at what price or you could easily find yourself stuck with six pets of the wrong breed that looked underpriced but actually were not.
Or, if your really hate dealing with the AH or grinding out gold, consider the suggestion at the end of Step 2 above about pet leveling and selling your extra flawless battle stones.
I know I may be encouraging potential trading partners to become AH competitors here, but it makes me very uncomfortable to see people suggesting they are spending money they probably shouldn't. Besides, I personally love playing the AH maybe even more than I love collecting pets and welcome any other "dual addicts" to the fold

Good luck!