Welcome on the game! I'll try my best to answer your questions, but understand that English isn't my first language.
I can't figure out how to earn floryns.
There are many ways to earn floryns. You can earn some through explorations, through the Glade (go in "Games", than click on "Glade", you can earn one gift per day), and through the Gothi-Memory (go in "Games", than click on "Gothi-Memory", it's a little memory game you can do if you want)
I can't use potions on my pets.
Potions can only be use when your pets are sick/hurt! So it is normal you can't use your potions if they health is perfect.
Don't worry, it will tell you if you need to use a potion on your creature.
I can't understand what the crystal shards are for.
Crystal shards can be used in the mine (click on "Adopt", than on "Gothimorphosis, than on "The Mine") Here, you can randomly adopt a Gothimorphosis, which are a mix of two creatures.
Here is an example:
Here is a Gothimorphosis
The pattern on the creature is a mix between those 2 creatures :
I can't understand how explorations work exactly.
To do explorations, click on one of your pets. On your creature's page, you have some options in a green bar. There, you have "Explore". Click on the button "Explore" and a little map will appear.
This is a map of Gothicat World. In the image in spoiler, you can see there is a little arrow under the Cloud kingdom (which is named Elonia). This is because the creature I want to send in exploration is living in this area. If you pick the country your creature is living in, it will only take 10 seconds to explore. You can make your creature explore any country, but the farest it is from its home, the longer it will take, plus, it will use more energy of your creature.
With explorations, your creature can come back with nothing, or with treasures (like floryns, jewels, potions, and sometimes, even creatures), or they can even return sick or hurt! When they come back hurt, you only need to use potions to heal them.
I can't tell if I'm looking after my pets well or not.
There is no "good way" or "bad way" to take care of your creatures. It is up to every player.
But on your creatures' pages, you can see little gauges:
If health is low, the only thing you can do is make your creature go to sleep.
If the energy is low, make it eat.
If the well being is low, make it play.
You can also increase Energy and Well being gauges by giving treats to your creature. (but be careful! If you give more than 2 treats, your creature will become sick of indigestion!)
Your creatures cannot die, so if you don't want to, you're not obligated to take care of your pets.
I hope it helped you with some of your questions!
Have a nice day!