I would think similar rules for playing an outsider, or a monster (like a mind flayer) could be used for playing a more mundane animal, at least to get it some character levels in a profession.
Once it gets a few levels, it can take a standard feat, like linguist, and then bob's your uncle.
Gives you 3 languages straight up (and should thus grant ability to talk), and 1 int.
Linguist
You have studied languages and codes, gaining the following benefits:
Increase your Intelligence score by 1, to a maximum of 20.
You learn three languages of your choice.
You can ably create written ciphers. Others can't decipher a code you create unless you teach them, they succeed on an Intelligence check (DC equal to your Intelligence score + your proficiency bonus), or they use magic to decipher it.
The helm is a gateway drug.
That'll take at least 4th level, since RAW the only way to get a feat before then is to be a Variant Human. However, there are other options.
Assuming we can't use backgrounds (because that's too easy), going through the classes in order, the first one that will give you a language is a Cleric of Knowledge Domain. It feels kind of iffy that any god would not only accept such an abomination against nature but also make them a chosen instrument of their divine will, but I suppose if any god would, it would likely be a god of knowledge.
Next, as a technical qualifier, is Druid, which grants proficiency with the Druidic language. It has very few speakers, but it's something. Also, Wildshape could potentially be used to transform into something with better communicative abilities, such as cranium rats, which can communicate telepathically.
So long as we qualify the goal to be a language before 4th level, Fighter is next at 3rd level, with both Cavalier and Samurai granting the ability to learn either some sort of skill or a language. Either sounds fantastic with a deinonychus, but I think Cavalier is the more hilarious option, because it's explicitly made for mounted combat.
After that, we skip a couple until we get to Ranger, which may be my favorite option, because it gains it's language through Favored Enemy, or, in other words, it hates someone or a group so much that it learns a language out of pure spite. Honestly, this seems very fitting for the subject in question.
After that is Rogue, which like Druid has is a technical qualifier with Thieves' Cant, so it can only talk with shady individuals, and it takes 4 times longer to communicate an idea that usual. Then, at 3rd level, it can pick up 2 languages by becoming a Mastermind (which is hilariously ironic, considering it will likely have am incredibly low intelligence barring a considerable investment in Tomes of Clear Thought).
Then there's Sorcerer of the Storm Sorcery origin, which just gives you access to Primordial. So you can speak with elementals and people who know Primordial. So possibly only slightly better than 1st level Druid or Rogue.
Warlock doesn't seem to actually grant any actual speaking ability, but it's very good for telepathy and reading. So the most direct way to go is form a pact with a Great Old One to pick up at will telepathy at level 1. Then, at level 2, when you first learn invocations, you might as well pick up Eyes of the Rune Keeper, so you can read ALL the language. Then, at level 3, when you get to pick up a Pact Boon, Chain is a pretty good option, since you can pick up a familiar capable of speaking Common for you (or a Pseudodragon to extend your telepathic communication to 100 feet).
And then I guess that's it (I thought there might be something in Wizard, but the closest is Divination School gaining the ability to be able to read all languages at level 10).