The warm-up actually starts before the ride. If you have tacked the horse yourself then you should have taken a good look at him and noticed his mental and physical state. If not, then briefly look over the horse and note his expression, demeanor, as well as his legs and feet.Next quickly check the tack to be sure it is adjusted correctly.
Be sure to have some small interaction with your horse before you get on. It could be a relaxed pat, a lump of sugar or just a careful look to see his demeanor.
5-6 minutes of walk work should precede every ride (at least two laps of the arena). First a free walk on a long rein and then walk work such as; turn on the forehand leverage turns, shoulder in or half-pass can be done. If your horse is tight nervous or will not relax into the walk work then maybe he should longe a bit before the ride. If he is still tight and uncommitted to the walk then you can go rising trot briskly around the arena until he is ready to walk. Always best to start the ride in walk whenever possible so aim for that.
Next, should be some easy trot half stretch getting the horse to start moving and establishing connection and rhythm for 3-4 minutes. Transitions between gaits in the walk and trot should be done, you can warm-up the canter in the same way. Note any stiffness or balance issues and see if they match what is normal for the horse. If a new issue pops up, take note of that and try to address it in the bulk of your ride.
Next begin with lateral work leg yielding, shoulder in or haunches in, whatever your horse can do, but do transitions between and within gaits in your lateral work as well. All horses have a stiff and a hollow side and that is likely going to be reflected in the lateral work.
Now you are ready for the working part of the ride or for your lesson. This should include anything you felt from the warm up that needed to be addressed and then what you may want to introduce or practice. Always try to end on a good note so your horse remembers that the end of the ride is when he is compliant and obedient, not in the middle of a struggle or argument. On that note…be sure if you do introduce something new you do so knowing how to ask for what you want and that your horse is capable of complying.
Be sure when you are done to let your horse walk and relax a little before going to the barn. A cool down is good for him and good for you to gather your thoughts about your ride.