You will need to buy some watercolor paper. It is specifically designed to not warp like that, though if you use a lot of water, it will warp anyway. There are ways that professionals get around that warping problem by "prestretching" a large piece of paper. You'd have to google that and watch how it's done. It's kind of complicated.
I don't prestretch mine, but since I dont' use a ton of water, I usually don't have a lot of warpage. What is there, is just part of what makes the buyer know it's an original, and not a print.
Another way to avoid the warping is to buy watercolor paper on a "block", where the edges are glued down to disallow this.
You will start with your lightest areas, so you must really think ahead in doing watercolor. Rmembe to leave blank anything you want to be white, becasue that is how it's white; it's the white paper showoing through.
I use a test piece of the watercolor paper, so when I mix up some paint in the pallette, I test it first on the tester to see if it's too strong or what., then go to the painting.
Practice with first wetting the paper in a small area, with just plain, clean water, then loading your brush with your color and laying down some paint abutting up to the wet area and watch how the paint wicks into the wet area. This is called, "wet on wet" and is a great technique for doing soft things like backgrounds or skies or clouds.
Also, try laying down an area of thick color and then sprinkling table salt on it for a lovely effect.
Don't use much black at all. Remember to use the color system of using complimentary colors to create shadows, just as you would in oil or pastel painting. So, for an orange horse, shadows with be made by adding blue to the orange color. For a yellow horse, purple. Black is almost never used and only in tiny amounts.
Play with the paints to see the effects you can get.
I like the work you did. It's important to lay it out pretty firmly with pencil first. WEhn it is totally dried, you can erase all the pencil without damaging the paint.
Good luck and keep them coming!