Hello!
There are quite a few ways to get new pets.
You found the first one which is to
Adopt the monthly pets. Every month at exactly midnight CS time new litters are released for users to adopt, you can check the time on CS
Here.
The second which you also found is to check the
Free pets forum. Plenty of users host giveaways, adoption centers, or just informally give away pets. Definitely a place to check frequently.
The
Pound & Lost and Found is also a good place to find pets and items. Users may donate pets to the pound, and some make games out of donating really rare pets for users to find. Banned user's pets are sent to the pound as well. They open at random times throughout the day but it gives you a six hour countdown. The lost and found opens in 1 hour and 44 minutes.
I'd recommend reading the newbie guide as well, there are a bunch of helpful links
Here, where you can read about trading as that is also a major part of pet collecting on CS.
Once your pets start to grow you'll likely end up getting duplicates, which you can trade for missed outcomes, when that happens you can make a
Trading post, asking users for trades. or you can look through the monthly trading post
Here where users can post pets from the current month that they're looking for. I linked Junes but you cna always find the current one at the top of the
trading forumAnd finally you can make a post in the
new member forum introducing yourself. You'll find other new users there as well.