Standardbreds are often very sweet natured horses, but like all horses you need to be the leader. You need to tell him where to go and when.
You're in a bit of a bad position, because you won't really be able to learn to trot properly until the horse can trot calmly and properly, but he won't be able to learn to trot like that until you, or someone else, teach him. People don't ride unsuitable trots like a normal trot, if my horse were to take off at a crazy trot I wouldn't rise along merrily, I'd sit back and slow him down, and only when the trot was correct would I start really riding the trot.
Horses are expensive, and if your horse is young, and you are inexperienced, a trainer or instructor is often almost as important a cost as feed, or agistment. Really consider the suitability of you and this horse.
Also, remove the problems. If your horse runs to your family tell your family to go away. Work on control, get it perfect. Get him weaving around things, doing tight circles, stopping immediately, serpentines etc at a walk. Never let him choose his own way or direction. Then do two or three strides of trot and then walk and slowly work up to more. Work on a 20 or 30 metre circle, get him used to be bending around the leg. Standardbreds are very "straight" horses, but you're going to have more control when you teach him to bend around your leg and keep on a circle. Until you have a really good trot where you can weave and bend, go fast and slow, do serpentines etc, don't even consider cantering.