LOTS of it is lighting. So think of lighting as the thing you need to learn about first, and composition and so forth can come next. You can crop a shot in a paint program, but you can't alter the lighting!
As others have pointed out, making your own lightbox is really easy. I did it myself with a moving box I had lying around, a sheet of white posterboard from the dollar store, some white tissue paper from the dollar store, and some tape left over from Christmas presents. So, cost, $2.00. Oh, and I used a box cutter I already had, but you could even just use scissors. I expected it to be a big pain, but it took me about twenty minutes and was super-easy and very much worth it for the little jewelry things I was photographing that day. I mostly take pics of larger stuff like big animal skulls and my silly cats, so light boxes aren't real useful to me, but for smaller stuff like ponies they are ideal.
To get good pictures, you should get acquainted with your camera. I have a cheap digital but I get okay results because I have learned its settings really well. Invest in a tripod if you can, you can get them for pretty cheap if you shop around online or you can even find them at like Best Buy. If you can't, you'll need to set your camera up on something elevated like a stack of books or another box. A steady foundation is really important because it allows you to use the higher-quality settings on your camera without taking a blurry picture.
See if your camera has a macro setting, and if it does and you are shooting something that is very close-up, use it. See if it helps you get a clearer picture. If you are in the right range, not too close and not too far off, you will probably find it helps loads. My old camera had a KILLER macro setting. My new camera is more expensive, but the macro setting is not as good. It is still necessary for shooting anything closer than about three feet, and I take my pony pics from about two to two and a half feet away.
The shake from pushing the button can make things blurry. See if your camera has a delay feature; mine has a two-second delay preset that I use because my camera has no input for a remote trigger. It's as good as the remote trigger would be and keeps my shaky hands from jiggling things around.
Set your camera to the lowest ISO you can. If the camera is stable, it won't cause blurriness, and you'll be getting a lot more detail and less noise.
Rule #1 of lighting: don't use your flash. It's too harsh, it washes out and flattens images, and while it's okay in a pinch for a quick in progress shot if you don't want to break out the whole shebang, it's not good for portrait type shots.
Natural light during a cloudy or partly cloudy day is ideal, since direct sun creates really sharp shadows. A room with lots of windows is good if you can't shoot outdoors. My house has no good windows, so I use two gooseneck lamps. Cheesecloth can be used to soften the lights, or tissue paper taped over them if they are low-temp. Your camera might have a setting to adjust for different light sources, which affect the color of the shots. If so, experiment and find the one that works best for the light you are using and the pony you are shooting.
You want lights to either side at about a 45-degree angle (halfway between right in front of the subject and straight out to the side). Experiment with the angle. One light is not usually enough, you need another to light the shadows some, to show detail. One can be brighter than the other; mine are.
Outside, you can make a crude reflector by taking a white sheet of posterboard, or even a white sheet, and setting it on the shadow side to reflect some of the sunlight into the shadowed area. If you are light-skinned, in a pinch, even your HAND held just off frame can reflect light onto a pony's legs or face or whatever spot needs just a touch of light. White tee shirts. Stuff like that.
Pick an uncomplicated background color, or one that compliments your pony. No bold patterns. For a dark pony use white, for a light pony, use a dark fabric, or use grey for any pony. I have a bunch of swaths of velvet about 2 yards each that I pick up each year, about two a year, at the post-holiday sales at fabric stores or with coupons. I have them in neutral gray, black, white, and a bunch of jewel tones I use behind my cats and boxes and so forth, and sometimes my ponies. I hang them from binder clips slung over nails driven into my studio wall. You can clip the backdrop to a bookshelf and drape it down onto a small table or seat of a chair. You can clip it or drape it over a chair, even. I used to use all my black velvet clothes safety-pinned together! Skirt over the back of the chair.
Small chairs can be moved outside. Shoot in front of a brick or fieldstone wall, a stucco wall, a whitewashed wall, if your area has houses built out of that stuff. Bookshelves can look nice, depending on the pony. That can look nice. Get creative with it!
I do all this on a budget of basically nothing; I have no money for specialized equipment, beyond the camera and tripod and backdrops. I have also never had a day of photography training (and seriously, it shows, because I am good enough to take pics of my own work, but nowhere near professional) and in fact avoided it for years because I didn't like it at all -- and I still mostly don't, though it can be fun. I still get good results. You can too, if you fiddle around with it and are patient with yourself. You will get better at it as you go along!
Your camera probably has something called "digital zoom" which is where it goes from being a lens-zoom, which is mechanical, to a digital enlargement of the image coming into the camera, which is not mechanical. My camera has a 10x zoom, but past 4x it's digital. I NEVER use the digital zoom, because it always results in severe loss of quality every time. Take it wider and crop it later. Looks better. Seriously, unless you have a crazy good camera, just skip the digital zoom.
Play with angle. Ponies can look very different depending on how you shoot them. From below, they look different than they do from above. Different poses look better at different angles. Accessories like flowers and such can be GREAT, but don't overwhelm the shoot with them.
It's like anything else, it's a skill that needs practice. It sounds way more complicated and way less fun than it is, even to someone like me, who is not terribly fond of it in the first place. Most of it is experimenting, fooling around, playing.
It helps to look at pictures people have taken that you think are really good and dissecting what you like about them and what makes them "work." Where is the light coming from? What color is the light? The backdrop? What angle was the camera at? I also haunt places like Etsy and look at product pictures people have taken and the creative stuff they've done, and how they lit and set up the shots. SO HELPFUL. Though my bank account doesn't like the inevitable results of finding all sorts of awesome stuff.
Also, make sure the area you are shooting in is free of cats. They are attracted to photo shoots like reverse paparazzi and will get all up in your biz while you are attempting to document ponyawesome. This can be adorable, but you don't want them getting into the habit or there will be a mishap involving red candle wax and your white velvet backdrop and there will be tears and recriminations and much gnashing of teeth. Be smarter than me and make sure they can't interrupt!
I hope that is helpful! I taught myself to take acceptable pictures of my work in less than six months of doing it maybe every other week, and I was really super-halfhearted about it. I am confident that you can learn to do it well enough to make yourself happy! I took lots of pics of cool stuff I had around the house (skulls, neat old books, jars and bottles) just to practice lighting and shooting different stuff.
And don't be shy about asking for feedback on your pictures, not just the pony, when you post them. People here are SO NICE and want to help, and will try to give you helpful tips about what you did well and what you could do better if you just ask them to and let them know you are open to constructive criticism in that regard.
Welcome to the MLP Arena! 
Guests: 857
Hidden: 0
Users: 5
