Congrats :) Welcome back to the world of riding. English is an excellent place to start, it will give you an excellent foundation to do whatever else you want.
For your daughter; of the many kids I've taught beginning lessons to not a single one of them has liked trotting at first. Your coach sounds like he was teaching you sitting, not posting, which is good-- posting can be a shortcut to learning to like the trot and being able to "move on" to other things, but I honestly think learning to sit the trot is a must before you truly CAN move on. In other words, encourage her to go back and keep learning. She will learn to like it eventually even though her (and your) backside will be sore for a few weeks or even possibly months. It's well worth it in the long run. But if she doesn't stick it out she'll never move on from not liking the trot and that could sour her on riding completely. Trotting pleasantly isn't a gift, it has to be earned :)
Enough with the sentimental Lifetime movie horsemanship from me for now, but one last thing-- even if your coach doesn't want you to buy boots just yet, please don't go and ride in sneakers! Go to Wal-Mart or somewhere and get a cheap rubber pair of rain boots with a 1/2 to one-inch heel for $20 and wear those. Any shoes with a tongue, laces, no heel and a low ankle (like most every-day-wear shoes out there) are extremely dangerous to ride in if you don't know what you're doing, and even if you do. If you come off the horse your foot is much more likely to get caught in the stirrup and you could be dragged.
State Line Tack also has cheap rubber English-style field boots on sale for about $20-$30 right now so you would even look the part with those :)