Well, welcome to the madhouse.
I'm going to offer a brief reality check here, things you'll find out on your own as you proceed into our little dark forest, but maybe more timely delivered now.
Using miles and points for travel is terrific and can add a lot of value to your life, but as with all things the devil is in the details.
Airlines make seats available for mileage redemption when their computers (which run very sophisticated, complicated and very secret programs) tell them that they will make more money by offering "award" seats than by waiting for the seats to sell for cash money.
For travel to destinations like Orlando in peak periods like December (not just kids going to amusement parks but also people going on Caribbean cruises) there are so many people lining up to purchase seats for cash (and lots of it) that it's going to be a real longshot to get seats for miles, especially multiple seats. I won't say "impossible" because nothing is, but the odds are hugely against you.
Which doesn't mean you should give up on accumulating miles, far from it, but learning how and when to use them, as opposed to using cash, is one of the real lessons to be gained by participating on this site. Don't be bashful or reluctant to ask questions or float ideas; we've all done it and have been in your shoes at some point.
Now regarding your specific circumstances, I'll offer a couple of ideas for you to pursue.
First, you don't say where in the "Pacific Northwest" you are, but if it's anywhere near Seattle, Spokane or Portland, you're going to discover that United Airlines isn't a major player anymore around here. (Their corporate HQ used to be in Seattle, go figure.) Instead, Alaska and Delta have staked out the PNW for themselves, and the former partners are now engaged in a death match to garner traffic and advantage. Alaska still has the edge, which they're trying to extend through purchase of Virgin America.
Second, the Alaska and Delta mileage programs are quite different, with different earning and use policies. In some cases one airline might have an advantage over another, but you won't know which is best for you without doing some research. One of the most important things to look at is an airline's partnerships. You can redeem Delta miles for travel on airlines like KLM or Air France or Virgin Atlantic, or redeem Alaska miles for a much wider range of partners including American Airlines and numerous overseas carriers. So if your mileage program airline doesn't fly somewhere you want to go, chances are one of their partners does, giving your miles more utility.
Third and possibly most important for your present case, airline branded credit cards have very different perks and policies. Which brings me back to Alaska.
In addition to a decent sign-up bonus, Alaska Airlines' Visa card (Bank of America) gives you an annual (electronic) "companion certificate" which conveys a $99 fare ($121 after taxes) for any Alaska Airlines trip where the "companion" is paying the going rate. So for example, say you're wanting to fly from Portland to Orlando in December. One ticket (you) costs, say $500 for a round trip (just guessing.) The companion (your child) flies the same route for $121, meaning the total for two is $621, or $310 per person. That's a pretty significant savings, and you can fly on any flights that have two open seats. The credit card has an annual fee of $75, but as you can see that's extinguished by the use of the companion certificate, and then some.
So my suggestion would be to do the homework, but consider the Alaska Airlines option. Not only is their mileage program one of the best, the credit card has a lot of perks, not the least of which is the companion fare.
Again, welcome!