Welcome to the forum.
The first thing I'd suggest, is taking a bit of time to learn about basic exposure. There are three things that make up an exposure...the shutter speed (length of time), the aperture (the size of the hole in the lens) and the ISO (the sensitivity of the sensor).
The amount of light that you have will affect what settings you need to make an exposure. Each of the three settings also has an effect on the photo.
The camera has a built-in meter that reads the reflected light and gives you the settings...when you are in any of the auto modes (anything but M).
The way that motion is recorded, has to do with the shutter speed. The faster (shorter) the shutter speed, the more you can freeze movement. The longer it gets, the more chance of blurry photos. So it would be reasonable that to freeze the motion of horse & rider, you will want to use a fast shutter speed. Just how fast will depends on a few things...but I'd suggest that something like 1/125 or 1/250 would be a good place to start.
That's well and good when you are outside and there is plenty of light. But when you go inside, there is much less light...and 1/125 may not be a long enough time to let in the proper amount of light to get a well exposed photo. The settings are all connected...so if you can open the aperture to a larger size, then more light can get in and you can use a faster shutter speed. The aperture is represented in F numbers. The lower the number, the bigger the aperture. If you notice what your camera is giving you when inside, it's probably already at the maximum aperture, which is somewhere between F4 and F5.6, depending on where the lens is zoomed to.
So since the lens is already at the maximum, the shutter speed that you get, will be the fastest you can get. But let's not forget the third setting...ISO. As you raise the ISO, the shutter speed can get faster. The trade off is digital noise. Shooting at ISO 1600 may mean that your images are quite noisy...but that may be better than blurry.
So there you go, when in darker situations, make sure that your lens is at the largest aperture (lowest F number) and turn up the ISO if your shutter speed still isn't fast enough. In some situations, it may still not be enough...and in that case...you will just have to get shots when the subject isn't moving as much.
Another option would be to add your own light (flash etc) but that's probably not a good idea when shooting horses.
Now, if you have noticed, one problem is the limitation of the lens and it's max aperture. A 'Fast' lens is one that has a large maximum aperture. Some zoom lenses go as large as F2.8 and some non-zoom lenses go to F1.8, F1.4 or even F1.2.
A faster lens will certainly allow you to get a faster shutter speed when used at large apertures. The most common recommendation is the EF 50mm F1.8 lens. It's quite cheap but optically very good. If you are on a budget, this would be my recommendation. If you have more money to spend, then there are plenty of other options.