Shutter Speed Help Needed for Newbie Photographer?

Well, I actually can't really call myself a photographer as I barely got my first camera   I'm trying to learn the basics, but what i need help with is a very specific situation and I thought I'd make a thread about it. Before I begin I apologize for asking something like this without knowing some of the basics of the camera and photography in general.

First of all let me tell you what type of camera I got. It's a Sony RX10 Mark III.

Now, what I'm trying to shoot and can't seem to get right is this: My brother does lots of horse racing and I always go for cheering but I also promised him to take great photos of him as he's riding. I'm always shooting in shutter-priority burst because we need as many pics we can get so ultimately make selections at home on a bigger screen. Now there are no races because of the lockdown but he can still go and ride solo for practice and I can also practice taking his pictures. The main problem is that I don't know how to set the camera so I don't get motion blur especially around the horses legs as those are always moving very fast when sprinting. I would like to get nice and still pictures with no motion blurr as I shoot in a burst mode. I must also mention and important fact, that I always zoom in all the way (600mm) as the seats are kind of far from the actual racing field. If I zoom in all the way I can get very good photos of him and the horse both (in terms of composition) but I have to  follow him pretty fast with the camera so it's nice to just hold burst and focus only on framing him right and then select good photos at home.

Now, let me tell you what I tried so far: 

AUTO mode is my best friend since I don't know much about the other modes. In AUTO mode If I shoot, out of 30 pictures let's say, only a few will be without motion blur around the horse's body but usually there's always some around the faster moving legs.

I then moved on to try Scenery mode, more specifically Sports. This mode is pretty good, If I'm not zoomed in all the way I get great images. But if I do zoom in all the way I will still get some motion blur in some of the pictures (not as many as in AUTO though)

Finally, after doing a little research and learning about what shutter speed is, I tried playing around with it in manual or shutter-priority modes and the motion blur completely disappears if i go high enough with it but the quality of the image gets worse and worse as the shutter speed is set higher and higher.

So my question is just what can I do to shoot nice and still images of a fast-moving subject in burst mode and when zoomed all the way in?

PS. The scenery-sports mode is constantly changing its shutter speed depending on what the camera thinks is best, sometimes it goes really high (just not high enough) in this mode but the quality seems to remain good, unlike in manual or shutter-priority modes. But still with a little motion blur. I feel that it would be perfect if I could get a little higher shutter speed in this scenery-sports mode as the final quality seems great, but sadly this mode doesn't allow modifying the shutter speed manually.

Thank you and best regards to everyone!
Andy

You should start by switching to Manual Mode or Shutter Priority Mode. I would begin with a shutter speed of 1/1250 sec. and progress my way up to faster shutter speeds. If in Manual Mode, set the aperture as open as you can. That would be f-4 for the zoom length you are using. Set ISO to Auto. This will be the best you can do. The main issue is the higher the Shutter speed, the higher the ISO it will require and the small 1" sensor on that camera will lose it's efficiency with the higher ISO. It will begin to lose color fidelity, dynamic range and it will have increased noise. It is a good camera but it has limitations. That is why you see sports photographers using the rigs they do. They don't tote all that huge gear around for fun. It has a purpose. With that being said, you should be able to get some good images or good enough images. Another key is getting closer so you don't have to use the longest zoom setting you have. Getting closer will improve your images dramatically. Good luck and let us know how it goes. 

+1 on all of the above.

When I shoot jousting, I can freeze motion with a shutter speed of 1/1250s. But they're moving pretty slowly. Horse racing, you are likely going to need 1/4000s or faster (n.b. 1/4000 won't freeze a hummingbird's wings). I add this to help you set expectations.

Bursting is a generally bad idea for a long-term strategy. What we do is learn to anticipate, know what we want, and capture it. You're new to all this, so you have lots to learn. But you've chosen an activity that takes practice, knowledge and skill.

If you want to educate yourself, start with learning about manual mode. Auto or scene modes are not your friend, and will not meet your needs. But that camera will serve you well in learning about photography, and you can always rent more powerful gear when needed.

We all start from the beginning.

Very informative, thank you both very much!
Indeed there is a lot to be learned here, but it's so intriguing and fascinating. So many options and possibilities  I'll make sure to experiment with everything you have suggested!

Get out of 'modes' and learn how to use the camera. Practice, lots of practice. Right now I doubt there are any sporting events (local) where you could go take pictures (where attendees are allowed to bring cameras and take pictures). But you could practice on whatever is around where you can take some pictures of something/anything just to get better at framing shots, being able to set the camera to get proper exposures, and to practice figuring out settings for a variety of conditions.

I've done sports, mostly hockey. For sports you need to use fast shutter speeds. Use your meter to indicate if your camera settings are giving you an adequate amount of light for a proper exposure. I can manage with a slower shutter speed if/when absolutely necessary due to lousy lighting (and unable to use a flash at event level), but it took lots of practice to get good at it.

Get as close as possible. Think about vantage points. Think about what backgrounds you'd be getting (blurring images does NOT get rid of crap back there that makes for bright or colorful out of focus blobs).

Sports photographers do sometimes use bursts. But you need to learn how/when it will be effective. Continual bursts will just give you a bazillion lousy pictures to hunt through. Slow down and learn what to do instead of firing off tons of shots hoping something will turn out. Odds are against that working and it'd be better to learn what to do so you can have success with this.

I would like to add you may want to try a slow shutter speed and pan with the subject. This is more desirable at times because it will give a sense of motion and speed. Don't be afraid to experiment on cars, motorcycles or bicycles going down the road. It's digital so it's free. Play away.

Shutter speed controls motion blur.  You can read about it and the other elements of exposure by reading about the exposure triangle.  After you have familiarized yourself with that, then you should use manual mode for a while and get a feel for how things change with the three corners of the triangle.  After you can predict the ideal choices for exposure you won't be able to say you aren't a photographer.

G'day Andy

The other thing I would add for you ... asap get out into your local library and chase up a beginner's guide to photography (no offence meant here) and see illustrations from experienced people.  
ps- Op Shops also have many similar books - many from film camera days - but the camera / photo basics are identical to newer digital cameras

Hope this helps
Phil from the great land Downunder
www.flickr.com/photos/ozzie_traveller/sets/

Wonderful and informative suggestions, thank you all so much!