Feeding Time

Well honestly whats best is not to even use that free water you got at all. But as a new player starting out, you can't afford to buy the good food for your pet yet.

The better food, gives the pet loyalty to you, which increases the pets xp gain (growth rate) and fighting abilities improve.

Just feeding the pet water, don't add loyalty at all, I'm sorry to say.

A standard rule I have is carrying 500 water and 20-30 oils.

When I hatch a pet, I'll sit in town and feed him an oil 7 times, that usually gets it close to the 500-900 marker (can't recall how high exactly, but it's happy pet time).
I then use my speed buff as my food timer, and feed it a water every time I rebuff.
There are occasions I have noticed where you may have to feed it sooner: pet taking too much damage (under 1/2 life), multiple enemies beating on it, they get upset and want food sooner from these kind of events, it seems. When the pet icon goes green, it wants food, a nice full color photo of your pet is the best scenario.

There is also the death of a pet, 1 oil brings it back to before it died immediately, and you then can resume the water rotation.

Just my method, hope you find it helpful.

The different levels if you don't feed your pet are:

Clear - Full - feeding gives 100% - 0 loyalty lost in 5 min
Green - Satisfied - feeding gives 100% - 1 loyalty lost in 5 min
Yellow - Peckish - feeding gives 90% - 5 loyalty lost in 5 min
Pink - Hungry - feeding gives 80% - 15 loyalty lost in 5 min
Pink - Hungry - feeding gives 80% - 15 loyalty lost in 5 min
Red - Starving - feeding gives 60% - 50 loyalty lost in 5 min
Red - Starving - feeding gives 60% - 50 loyalty lost in 5 min
Dark Red - Famished - feeding gives 30% - 100 loyalty lost in 5 min

When you feed your pet, it goes up one color (that is, Hungry and Starving are not repeated as they are on the way down). Note that if your pet's hunger level is not clear or green, feeding it expensive food will result in less than 100% of the expected loyalty gain. So use water to bring its hunger level up to clear or green, then feed it the expensive food to gain 100% of the loyalty gain for the food.

Likewise, if your pet's loyalty is not maxed (or however high above 500 you set as your target), you should feed it when it becomes green to maximize the loyalty gain per unit of food.

Once your pet's loyalty is maxed, your best strategy is not to feed it until the portrait goes pink. When it goes green, it doesn't lose any loyalty. So you're using a water to gain nothing. When it goes to yellow it will lose 1 loyalty and should gain 9 for a water. But since the loyalty is 998, you're using a water to gain just 1 loyalty.

When it first goes to pink it loses 5 loyalty (down to 993 if you started at max) and should gain 8 for a water. So you're still wasting 2 loyalty points if it started max.

5 minutes later it'll lose 15 loyalty (still pink, down to 978) and gain 8 for a water.

So the break-even point is in between the two pink levels. You can feed your pet about once every 15-20 min and keep it maxed/near max. If it started at max, you should feed it once every 20 min. If it started just a little under max (a little space visible at the end of the yellow), then feed it the instant it becomes pink. That'll let you maintain the pet at near-max using the minimum amount of water possible.