I'd say the free adoption board would still be the best place if you are looking to just get more pets in your trade groups in general. I would suggest though to focus more on adoption centers that allow users to send trades daily/weekly and making yourself a short list of places to revisit that you are able to get pets from reliably. Threads that consist of big giveaways, clearly out whole accounts, etc, are great for getting better pets, but there is also a lot more competition for those pets and its more likely the trade will get auto-cancelled because someone else had sent in a trade for what you asked for before you. If you are able to as well, keeping your requests to smaller groups of pets will also help with not getting autocancelled as often, since if you ask for 50 pets and even just one of those was asked for by someone else, the whole trade gets autocancelled. If an adoption center allows it, multiple smaller requests are more likely to get accepted.
We also have 4 big events every year that are great for getting more pets as a newbie. Our next one is the Easter event which should start late march/early april. Every day of the event free pets or items are released, and while the event is active you can collect tokens while browsing the site, you can then trade those tokens in on the event page for event pets and items. If you want an idea for what gets released you can look at past year's events, easter has a lot of bunnies and general pastels, flowers, and egg themes, etc, as would be expected. The free event pets always end up pretty common, but the pets you can only get from trading in tokens are essentially guaranteed to be at least Extremely Uncommon or Rare (the more popular are the more common they are since rarity is determined by how many are owned after the event, so its never 100% certain what the rarity will be). Events do require a decent time commitment, tokens have a 20 min cooldown so once you collect one you have to wait 20 min to get another, and there is also a cap on total tokens you can collect by the end of the event (usually ~300 total). So keep that in mind if you plan to participate. You can trade for token from other users though, or trade them your extras, and the trading scene tends to get way more active during events so its a good time to participate.
Last thing I thought I'd mention is that if you like to draw art at all, we have a pretty big art community here and a lot of people like to draw commissions for people, design characters, etc, and can then trade/sell those for CS pets, C$, etc. to get extra trading stock. Even if you don't like to draw that often or do anything that in depth, if you are just looking for extra pets for trading you can probably find someone interested in sending you some of their spare pets for art of their characters. There is an
art shops board and
character sales board, as well as the whole of the oekaki where you can look at that if you are interested in that sort of thing. There is also a coding shops board for bbcode coding, and you can sell writing as well, but drawn art/characters is definitely the most popular/easiest to sell here.