There are 2 main aspects: the first is related to the composition and what you want to transmit to the viewer with your image. This is extremely important and I have seen photographers that excelled in the technical department but their photos lacked a certain mood, feeling etc. Usually the composition is vital for this - it’s something that can be learned and it’s really worth spending time to consider the shot and plan it in your mind before you press the button.
The second thing is the technical aspect of things. You must learn how to get out of the automatic mode and use the camera to create the style and image that you visualised before pressing the shutter.
For example, in your portrait of the horse, I’d say maybe from a composition point of view, it’s ok but if you had chosen a wider aperture, for example, you’d have really concentrated on the face and his eyes , blurring the distractions of the background. (Wider aperture- I mean a low f number). With a low f number you achieve a shallower depth of field, allowing your viewers to look at what’s important in the image- your main subject, the horse, and not let their eyes wander aimlessly at the fence, rails in the background etc. The colours are also a bit over/saturated in that particular example that you chose.
The second image is much better - the composition is good, you have some leading lines to keep your viewer interested in the image. Applying b&w also makes it pop a bit more. In this case you didn’t need a shallow depth of field, so it worked.
If your camera is 10 yr old, I’d imagine that the RAW files are not that big, so always use the option of shooting in both raw and jpeg (there’s a setting that allows you to capture 2 files for each image). Have a raw file for all your images- working later with JPEG’s is actually not that great. You can make your picks and discard all images that you’re not happy with- so don’t worry about space on the hard drive, concentrate on the highest quality of your pictures that you can achieve. When I shoot, I don’t know in advance- I’m going to edit this - I’m not going to edit that etc. All images have the potential to make it through my selection so it’s important to have the best file quality.
When you look at a batch of images and discard some of them, include another step in the process: think about why you are not choosing the image: what’s missing? It can be wrong focus, maybe the image is blurred, etc. This will also give you a clue on where you need to improve.
Hope I’ve given you a few ideas. Definitely play more with the three main elements: f number, shutter speed and ISO. They are the key for achieving amazing results!