Just post an ad for your pet, talking about its personality, its likes and dislikes, and what kind of pet-date it's looking for.
Next, browse though the current ads (i.e. if you have a female pet, click on "Find a Date!" under the "Girls Looking for Guys" section) and see if you find any dates your pet might be interested in.
Once you've found a pet you're interested in, neomail the owner! This will hopefully start a date! Have fun! =)