After defeating the first god,Hyperion, you will have access to your very first pet! The mouse, located under Fight->Pets.
Pets will not do a lot for you early on, though in general they increase the rebirth multiplier by 0.01% per stat. Example: 4 Battle= 0.04% more battlerebirth multiplier.
This doesn't sound like a lot at first, but it will come in very handy later on.
So how do you increase those stats? Well whenever your pet levels up there is some fancy math stuff going on (hover over the Total growth stat to see the formulars) and that will increase the stats of your pet.
Growth is affected by the food you give to your pet, you can do that whenever the so called hungerbar is below 75%, always feed your pet when possible. When feeding it you have various food types and sauces to choose from, don't bother with the food types yet, though the sauces are all free and add different things to your growth, you can decide on your own what sauce you want to use. I personally went for sour sauce and mayonaise alternately.
To get your pet to get experience you need to create some custom clones first, do that with clicking on the Create Clones tab. there you can change the stats of the custom clones, make sure their stats are slightly below the ones of your pet.
You can also set the amount it shall fight, set that whatever you want. Make sure your pet is not losing HP when fighting clones, if it does, lower the battle/physical stat of your clones. That will slightly reduce the xp per clone but greatly increase the xp over time, trust me.
To get your pet to fight these clones you need to distribute them to your pet. To do that you need to go back to the pet-mainscreen. On the top right it will say distribute, clicking that will move all your idling Clones to your pet to fight it. don't move more than 50-100 early on, it's not worth it. Reclaiming will simply take back all Clones you distributed before that haven't yet been defeated by your mighty mighty mouse.
That should be it on the topic pets for now, I'll add a GP guide within the next few days and afterwards a guide on monuments or vice versa, thanks for all the positive support, leave comments, suggestions, requests, corrections, or simply advises or recommendations in the section down below. If you find any kind of grammatic/spelling or phrasing mistake, just tell me, I will correct it ASAP.