While in battle I use the heart when I can to heal it but that's not enough I need it to be completely healed. How do I do that?

You can use elixirs in battle to heal pets more. The light and dark proficiency trees also have some healing abilities, but nothing big.

Where do I find these tree's? I am only new and can only explore so little. Also what if I am not in battle then how do I heal my pet?

If you're not in battle you can either use potions and elixirs (can be rewards from missions, found while exploring, or bought from the town market). You can also visit the plaza in town and she will heal your pets to full for gold.

Okay I am in a battle right now and It won't let me use an elixir. I clicked the red arrow thing, switched to my items using the button on the bottom of the pop up and it wont work

To use an item in battle, you have to click on the item, and then click on the pet that you want to use it on. Are you clicking on the pet as well? If it's done right, a little potion icon should show up by the pet name to indicate that a item will be used on it during the next turn.

Okay Got it! Thanks.... What about capturing new pets? I bought a snare and I will find out how to use that but is that the only way to capture the pets? Also In the tutorial it was talking about obtaining something that you can take to the lab and create your own creature or something like that. And one more thing. How do you obtain more gold?

Using traps is kind of similar. You'll use the little bag looking icon to see what traps you have, click on a trap and then click on the pet you want to catch. It'll show a little net icon over that pet to indicate you're going to use an item on it in the next round.

And the item it was talking about is an essence. You can get these from lots of different places (games, battle, rewards, nurturing, just about anywhere) but the chance of getting them is pretty low. You can go to the generators and use an essence to create a pet where you select all the mutations, genes, and colors.

Essences can be either regular, enhanced, or mystical. Regular gives you 2 genepoints (so either a 1 vis pet or a 2 carry pet). Enhanced gives you 4 gene points. Mystical also gives you 4 gene points, but unlike the other two it is very very accurate with colors. Regular essences will almost always have some swing to their colors (so a pet you wanted white might turn out a light grey) Enhanced essences also have some color swing, but not as much as regular ones.

Also, while you can find regular and enhanced just playing around, mythical essences are only available from the diamond shop (you can buy regular and enhanced with diamonds as well)

And then what about obtaining gold?

Oh man there are so many ways to make gold I'm going to link you to a great guide here rather than typing out the whole thing.

I found at lower levels the easiest way to make gold was just to battle a bunch and sell whatever drops. You can also have a pet or two doing missions all the time, which is a good way to earn steady gold.