There is a major flaw in other answers: they might surprise some humans and managed to kill a few, but in the long run (read: in days or 2-3 years) humans will dominate again.
What do you think will happen if you see mice destroying crops, pets attacking owners, insects purposefully spreading diseases, large animals stampeding and destroying houses?
The answer is: humans will hunt them into extinction (or at least until they can't do it again). Humans will pesticide their crops with stronger pesticide, kill or disown their pets, wear hazmat suit to prevent insect-spread disease (like how we wear mask during pandemic), and shoot those big (read: k-selected) stampeding animals.
Even if those big-stampeding animals learned military tactics (e.g.: pincer attack, cutting supplies, hit-and-run tactics, form heavy hammer in large number), they're still nothing against tanks, battleships, artilleries, or any other heavy weapons. If everything else fail, humans still have their N option.
Here is my assumptions:
- Human tech does not advance in those 300-400 years in your plot (equal to today's tech)
- All animals (only from kingdom animalia, not including plantae/plants, fungi/shrooms, single-celled organism) gained sentience equal to human (not all individuals are equally smart, some individuals might have their own special 'talents' or specializations)
- All those newly gained sentience developed from zero (no dogs/tigers can be handwaved to immediately have intelligence+knowledge+skills of submarine engineer or surgeon or fighter jet pilots). So when a dog get its sentience 1 year ago, it's only as smart as 1 years old infant.
- Human does not have any moral hesitation to kill animals who they believe to threaten them (as how we don't have moral hesitation to kill centipedes or roaches since we think they threaten us, but we still have moral hesitation to kill pet cats or dogs since we don't think they threaten us)
- The animal's objective is to remove human from "Ultimate Apex Animal"
- Not by ONLY becoming larger number than human (I believe ants/bees/other insects might be more numerous than humans)
- Not by ONLY being able to eat humans (sharks and piranhas might eat few humans each year, but they're not "Ultimate Apex Animal")
- BUT by ruling the world:
- When a significantly high-ranking animal leaders decide to do something, it CANNOT be foiled/interfered by human by force (same as how if human leader decided to lock down a city because of some virus, no animal can "veto" such decision by military force or economy threat or nuclear threat)
- No human should be able to recover their "Ultimate Apex Animal" (at least not in foreseeable future, e.g.: 100-500 years)
Under such assumptions, here is the possible scenarios and their requirements. I will write it in "game guide" or "walkthrough".
A. Animals right ending
Story:
Animals become acknowledged same right as human (e.g.: there will be Universal Declarations of Sentient Life Rights / UDSLR). UDSLR includes right to live (not killed), economic right (right to buy/sell and ownership), right not to be enslaved, etc...
Walkthrough:
This is the easiest ending: animals have sentience, they just need to establish some kind of communication protocol (read: language) to communicate with each other and with human. After that they will have to prove that they're also like human (can create value system, can create legal government, can appreciate art, etc.). After that human will naturally protect their right (not immediately of course, but after many grueling battles-slaughters-armistices).
B. Animals nation ending
Story:
Animals established their own nation, with their own respectable military and economic might; that human can not attack it without feeling that their attack is 'not worth it'.
Walkthrough:
Do the same with above A ending. But after that try to amass large number of animals in empty lands (that is not owned by humans yet). Try not to have any open conflicts with humans: when humans decided to kill the animal factions then your playthrough is basically a game over (read: extinctions).
Try to produce excess food there (farming), establish social hierarchy, develop laws, weapons, etc. and accept animal immigrants from human nations. Those animal immigrants should ideally already absorbed human "intellectual properties" that the humans might give away for free (e.g.: calendar, simple mechanics, food preservation technique) or even highly critical IP (e.g.: medicine, surgery, etc.).
Even then their strength will still only on par of today's third world country. So trying to conquer the world and space is still very way off...
C. Conqueror of the earth ending
Story: animals established their goal listed above.
Walkthrough: this is the hardest one, that previous endings feels like child's play compared to this one.
While the animals developed sentience and intelligence, no humans should know about it. Or if they know it, then they must not know that ALL animals have intelligence (not just some). Fact that billions of creature on earth gained sentience HAVE to be kept away from human. That include all animals that are being experimented by humans to animals intelligences.
- If this step is not cleared: humans might feel threatened by animals and might create a few facility/zone that is free from animals. Those facilities will be human's last (and successful) fort when retaking their "Ultimate Apex Animal" status. From those facilities humans can develop weapons (e.g.: flamethrower to burn houses infested by roaches, poison gas, or even nuclear-based weapon when they think a continent have been overrun/dominated by 'evil' animals)
Animals faction must be able to communicate (preferably instantly, reliably, and confidentially) with each other over very long distance. Carrier pigeon is not enough (there's a long delay). I'm talking about satellite/cellular communication level. This might be handwaved (e.g.: the animals found out a frequency that cannot be heard by humans, and that frequency has very long range that a whisper is heard planet-wide, and that frequency is undetectable by humans even with latest radio equipment).
Animals faction must establish an internal absolute order. That means there's a mastermind animal that can order other animals to execute suicide missions (e.g.: attacking humans IS PRACTICALLY a suicide mission). There is also no animals that want to usurp those 'mastermind animal': in fictions this is often called 'hive mind'.
With long range comms and absolute order, animals establish a planet wide coordination (if the long range comms is not planet wide yet they might just add some 'repeater'/'relay' animals)
Animals faction will have to attack all at once (no delay between their attacks), then secure all humans' critical section (e.g.: military facility, economic production, communication facilities, governments to prevent humans from coordinating at national scale). This attack have to happen at the same time all over the planet (hence two points above), and all have to be able to successfully take over all human civilizations.
- The attack itself can be quite simple: mice/mosquito induced plague, rampaging bears/lions/elephants, suicide bombing by birds/insects into gas stations, military dogs or monkeys infiltrating nuclear facilities and setting them off
- The real hard part is to maintain this dominance over humans:
- If only some attacks are successful, humans might regroup in other country and launch punitive force against the animals faction. This punitive force already drilled on how to kill animals and already learned animals' tricks, armed with specific anti-animal weapons (e.g.: gas, sound frequency that animal cannot stand)
- Remember that even if the animals is intelligent, they still do not have the facilities and the know-hows on how to create, operate, and effectively use advanced weaponry (e.g.: missiles, jets, explosives, nuclear, gunpowder, genetical engineering)
- The animals doesn't stand a chance at waging long asymmetric warfare (e.g.: guerilla tactics). Asymmetric warfare requires that the other side to be morally hesitant to exterminate the other side. Once the humans think that all animals are waging guerilla warfare against them they will kill all the animals.
D. Ruling from the shadow ending
Story: animals faction learned through simulated scenarios that it's impossible for complete takeover in ending C. So they threatened/blackmailed world's top peoples (e.g.: American government members, including all of their executive, legislative, judicative and all other branch) to do their bidding (e.g.: passing law that said killing animal is illegal, make movies and propagandas to love or worship your 'pets', etc.)
Walkthrough: to blackmail them successfully, animals faction can do one of the following:
- Get secret information (e.g.: politicians having affairs? get a mosquito or ant to film an unrefutable evidence; corruptions? those hair lices can testify at court)
- Threaten to assassinate (e.g.: mosquitos with knowledge that they carry deadly viruses can threaten to kill a politician or even all of his/her family if they don't comply)
- Divide et empera: tell two big opposing forces in human world that their enemies are going to attack or something to that effect. After humans are busy killing each other, the animal factions can try to further push their agenda.