Once your horse has ran a race, you can view results in a couple of different places. First, you can go to their page and find the Race History link on the side. This will show details about all races they have ran in, along with the notes they got in those races. This is a handy way to see what you need to adjust for the next race.
You can also view results for an entire track by going to the Results page in the Racing tab. This can be good if you want to keep an eye on competition. If you can see how those horses ran, and how yours compared, you can often learn a lot about how good (or bad) your horse is, or what you might need to adjust.