It would appear from this bit of code that you have register globals set on, not a good idea.
To change from GET to POST you will need to change your method in your form to POST, from GET and you will then have to change any variables
that are being referenced as $GET to $POST.
However in your case I think that you need to look at the manual link supplied above and change lines like this
<?php if(isset($selectedhorse)) {
to look like this
<?php if(isset($_POST['selectedhorse'])) {
as well as going through your code to see where your variables are getting assigned, for instance you use the $selectedhorse variable in your SQL query directly, again not a good idea. Take a look here to see the reasons why it's a bad idea.