Adult Beginners piano lessons
I'm a beginning adult piano student (or at least I was five years ago) and can relate to having learned the instrument as an adult with no prior music background.
I've always wanted to learn the piano as a child (my mother played a little), but never did. No point regretting what I cannot change.
I think the best thing that you can do to prepare yourself for the journey you're about to begin is to realize that it is, in fact, a journey. The journey is one which requires a strong commitment on your part to learn. Many adult students never make it past the first year of studies because they did not realize that making progress on the instrument requires daily practice. Just as you did not learn to read and write going to school once a week, it is virtually impossible to progress by practicing once a week. Most piano teachers recommend practicing from 30 to 60 minutes per day. Even if you can get 15 minutes of practice in on busy days, that is better than nothing. Practicing 3.5 hours, once a week, is not the same thing as 30 minutes per day. Concepts take time to settle in and be digested. It is not a test your can 'cram' for.
For many adults, this presents challenges, if they have to balance practice with a job with long hours, family commitments and other social obligations. However, the bottom line is that you must make time for it. If you want it bad enough, no matter how busy you are, you'll find the time.
Another thing to keep in mind is that the journey will be simultaneously gratifying and frustrating. That isn't meant to try to scare you, but there will be times where it doesn't seem like you're progressing. "Hey, I practiced for two hours straight and I still can't execute this passage properly!" Those words were likely said by every person in history who has tried to tame this beast. The learn is so minute, that you will not notice it over short periods of time. It's like watching paint dry or hair grow. However, over time (months and years), you'll be able to hear and see all of your hard work pay off. Accept the fact that this is not a ukulele and that you'll be strumming a song at the end of the afternoon.
The complexities (and beauty) of the instrument require specific skills. Each one must be practiced until you get it right. How long does it take? It takes as long as it takes. It could be days, weeks, months or perhaps even years (that is rare, though). Each of these skills builds upon one another and, over time, you'll develop an 'arsenal' of skills that will be at your disposal.
Find the right teacher. One you click with and one who will help you to achieve your goals. They will be your guide on this journey, offering you feedback, correcting your mistakes, listening with a discerning ear and providing you encouragement along the way. Keep the lines of communication open and let them know what you like and what you don't like. If, for example, your goal is to play jazz or pop and all they teach is classical, that is consideration that should be made before starting your lessons.
As for studying alongside your child, that question is best left to your teacher. To me (I'm not a teacher, by the way), it seems distracting to have two people in the lesson. You will both progress at different rates and your teacher's attention to either one of you will be split into two. I think I'd suggest separate lessons. Most piano teachers I've spoken two suggest that between the ages of 5-7 is an ideal time for a child to begin lessons. Any younger than that, and many have expressed concerns with attention span.
As for your piano, buy the best you can afford and upgrade when finances permit. I would very strongly advise against an unweighted keyboard, typically found with low-end digital pianos. They will not be replicating the feel of an acoustic piano very accurately and, as a result, you may develop some bad technique which may prove to be hard to unlearn, as you graduate to an acoustic.
Your question is so broad, I don't know where to end, but if you have any specific questions, I'm happen to entertain them.