Welcome to EpicSki, Emil.
Good answers to your question above, I think. Buying and adapting to new, high-performance boots is a big step toward becoming a better skier. I understand your concern!
I'll second LiquidFeet's advice to buy carefully, with the help of a good bootfitter. Boots are a big investment, and it is tempting to try to save a little money where you can. But the right boot, with the right fit and--most importantly--the right setup (adjusting all of the various fore-aft and lateral angles of the boot) is much more important when you step up to a high-performance, stiffer, snugger boot.
But there is not much I can think of that you need to do over the summer specifically to prepare yourself for new boots, short of getting yourself as fit as you can before the ski season. Getting in better shape is always a good thing. And of course--to LiquidFeet's other point--beware that increasing fitness, especially with weight loss or increasing leg strength (and muscle mass), can significantly alter the fit of your boots. If you're expecting big changes over the summer, perhaps you should wait until fall to get those boots. You will find some some good pre-season training ideas in the EpicSki "articles" section, here.
Get some good, thin socks that will not interfere with the precise fit of your boots. And buy a pair of "Cat Tracks"
or other similar covers to protect the soles when you walk around in the boots (the flat, smooth sole is a critical part of the boots' performance, and it wears quickly when you walk in them on anything but soft snow). They're a bit of a nuisance, but it is good to get into the habit of using sole protectors religiously--take them off only when you put your skis on (they fit, with some bulk, into a jacket pocket, or you can strap them into the upper buckles or "power straps" of your boots).
Before the season starts, and with those Cat Tracks on, you can wear the boots a little bit around the house, as jzamp suggests. But I wouldn't bother with going too far there. Get out on a bicycle, or hiking in the mountains, or playing tennis, or whatever else you do to get fit, strong, and quick, and enjoy the summer. Winter will come soon enough.
Metaphor has offered some good suggestions for early season on-snow. The new boots will feel "different," and the most important thing to do at first is simply to get used to them. (Along with that, expect a few more visits to the bootfitter to fine-tune the fit and adjustments.)
The next thing I encourage you to do--early on, if possible--is to find a good instructor and get some coaching and feedback. Those boots will work best with good technique, so it will be good to get some new focuses to help you develop there--and to make sure you aren't just practicing and ingraining bad habits. It will make skiing with your kids more enjoyable for yourself, too, if you have something you can focus on in your own skiing at the same time. And you may be able to help them improve more quickly as well. If you can't find, or afford, a good instructor, don't hesitate to have someone shoot some video of you and ask questions here at EpicSki.
You're heading for a big breakthrough if you do this right. High-performance boots and refined technique will make a huge difference. Have a great summer, and keep us posted. I look forward to hearing how it goes.