I've never trained a Standardbred, but here is what I have to share. You will need to learn about gait in three ways. First, you need to learn mentally what the timing and footfall pattern is so you can identify the gaits. Second, you'll need to learn how to recognize it watching the horse work (for example if you have someone video tape your riding). Then, you'll need to learn how to feel it in the saddle.
To start with, Standardbreds can pace or trot. Pace is a two-beat lateral gait (both legs on the same side of the horse set down at the same time) and at the other end of the spectrum, trot is a two-beat diagonal gait (opposite legs set down at the same time, such as right front and left rear). In between true lateral and true diagonal are the intermediate gaits of stepping pace, rack, running walk, fox trot, etc. The horse has genes that determine if he can pace or trot; then there are additional genes (walk modifiers) that give a gaited horse the ability to modify the pace or trot into a four-beat "intermediate" saddle gait.
I tried to type an example of the gait continuum below (I hope it is understandable):
Gait Continuum
|------------------------------------------|----------------------------|----------------------|-------------------------|
Pace Step pace Running Walk Fox Trot Trot
(Lateral) (Diagonal)
So, depending on your horse's genetics, he can potentially pace, trot, and maybe do a stepping pace or a rack. The genetics determine that. There is a fabulous book that is required reading (LOL). It is called Heavenly Gaits by Brenda Imus. She explains all the saddle gaits and where they fit on the continuum. The timing and footfall/support pattern is different, that what separates them. I highly recommend this book so you can learn what all the saddle gaits are; then you will be able to start identifying what your horse can do.
When watching the horse go through his paces, sometimes it is difficult to make sense of what gait he is doing, especially if he is doing it fast. You can use different color leg wraps, like blue on the left legs and orange on the right. Then, instead of figuring out which leg is doing what, you can see if both orange wraps go forward at the same time, the horse is pacing. If one orange and one blue go forward at the same time, the horse is trotting. If the horse is doing a four-beat gait and his head is nodding up and down in time with the gait, he is doing the flat walk or running walk. If he is doing a four-beat gait and his head is nodding from side to side, he is racking. If he is doing a four-beat gait without any headshake, he is saddling, a generic term for an ambling four-beat gait that doesn't have the overstride of the running walk.
The way that we train walkers (I'm an amateur so I don't know a whole lot) is that we start asking them to step up from a walk, and see what they do. Just like WTW said, take a deep seat and cluck. (Note: We're big on "clucking" in the gaited horse industry. 
My horse Gunslinger just waits for that one cluck that he knows he can kick in his afterburners and GO!) Then you can see what the horse offers you. Based on what he offers, you can do different things. If he paces for example, you might try doing a lot of circles. When the horse paces, his back is a little hollow, and the circle work helps supple the horse's back. So, you can make nice big circles and figure eights, and you can also spiral in and out. You can also take the horse over cavaletti to break up a pace, and you can work him up a hill, which shifts his weight and will sometimes put him into a running walk. If the horse trots, you can use other exercises to try to get him to ease into a four-beat gait.
There are two wonderful videos that Lonnie Kuehn has produced: Smoothing Out the Pacey Horse and Smoothing Out the Trotty Horse. These videos give natural exercises to help the horses find the right "gear" as we call it. It does make a difference in how you put pressure on your reins; however, I like staying out of the horse's mouth and letting the exercises shift his weight.
Once you find the gear that you like (like a step-pace), keep working the horse in that gear to build consistency. I don't trot my horses or pace them either. If they break into a trot or pace, I slow them down to where they have a four-beat gait and build consistency there. Don't let them speed up or slow down. When you get them in gear, you hold them there. The way one trainer I had put it: Don't change anything; just keep him going in that gear. You can see that in the GraveDigger video especially when he is riding on the road. The trainer put him in gear and just holds him there; the horse is totally consistent in his timing, like you wind up a clock and it just ticks in time.
Once you build consistency, the horse will go in that gear when you cluck him up. (Some Southern culture and lingo: "cluck him up" means to ask him for some speed, and getting him "clucked up" means you've got the horse going in his gear. I just love the lingo!! So, now you know if somebody asks you if got your horse clucked up, you know what clucked up means!!!!! If you are just doing a slow dog walk, you are NOT clucked up!!!!!)
With Gunslinger and Flashcube, we cluck to ask for speed and also voice train for "Show Walk" or "Show Gait." These are actually spotted saddle horse terms, with Show Walk being a flat-foot walk and Show Gait being a running walk or rack or whatever your horse does. I gently squeeze with my lower leg if I need to speed him up a little more. I don't have a cue for trot, so you will have to look another source as to how to best cue for trot. With the show horses, we actually take a hold of the bridle (take the slack out of the rein and put a little more pressure on the bit and raise our hands) when we ask for speed, because we're setting them up in collected frame. Once they collect themselves and get into gear, then we ease off the pressure and just keep a consistent feel in the mouth and just hold them where they are at.. This is what we do for Show Pleasure classes. If I'm riding a Country Pleasure class, then I let the horse stay on a slack rein, and his head is lower and his step isn't as high. This is more of what WTW described with her trail pleasure horses, she loosens the pressure on the reins to go forward because the horse is in a trail gait with a relaxed way of going (think of a typical Western pleasure horse with a relaxed head), whereas the show horses are collected (think of a saddleseat horse with a high head carriage and higher step).
Feeling gait in the saddle is another topic and there is so much more to say; but it is past my bedtime! Lee Zeigler does have a book out and she is considered very knowledgeable. All of Lonnie Keuhns videos are good, and there is a set of DVDs called Gaits of God by Brenda Imus. It is very expensive, but it does show examples of horses fox-trotting and doing running walk. I haven't looked at the whole set yet, but it might be worth putting on your Christmas list! Lonnie also has a shoeing video that explains about using weight on front and back to help get the horse in the right gear. Lonnie's videos are reasonably priced and well worth it; sometimes National Bridle Shop will run sales on her videos so you can get them for a few dollars off regular price.
So, take a deep seat, cluck him up, and see what you get from your horse!!!!!! Or as they say at the Celebration (and it sends tingles down my spine) "Okay riders, take a deeeeep seat and let 'em walk on! The music speeds up, the horses speed up, and it is just so exciting and fun!
Hope that helped, and I hope you enjoyed the culture and vocabulary of the South.
P.S. Can you tell that I just LOVE my gaited horses!!!!!!