I think it depends on how much diving she is going to do and it may take you a little while to judge that. Obviously, the more diving she does, the more sense it makes to buy all of the gear. If she is only going to dive 1 or 2 times a year, it's not worth the expense or the hassle of keeping the gear up, especially for items she'll soon outgrow.

If it were me, the very first thing I buy her would be a mask that fit her well. A mask that doesn't fit and leaks badly is one of the most irritating things on a dive. It's right up there with being cold, in my opinion. You might also want to get her a wetsuit that fits well, since it may be difficult to rent one that fits her well. Next thing I'd get would be a regulator set. Everything else would come later, for me, with tanks coming last.

You could also lean on your LDS for a while until you decide what to do. You could even rent gear from them and take it on vacation with you. At least you'd have more confidence that it had been properly maintained and you'd be able to work out any fit issues beforehand.

As far as traveling with gear goes, it's a hassle for sure. Whether or not I take mine, depends on what the plans are. On a dive trip or vacation where I'll be doing a fair amount of diving, I'm certainly taking all of my gear, except maybe tanks & weights. But if I'm only going to do one or 2 dives, like from a cruise boat or something, I won't take all of it usually. The first thing I grab is my mask and regulator, everything else is optional, especially if I'm going to warm waters.

Also, since you have little ones that are also getting (or soon to be) involved in diving, I encourage you to take your diving skills to the next level. Dive as often as you can in order to get as comfortable and as much experience as you can. I'd suggest going to at least the Rescue Diver level and to keep your CPR skills honed as well.

I agree with your daughter regarding boat dives. That's the only way to dive!