Would I be able to get what I need by soldering together 16 AA batteries in series? Or would internal resistance foil my plans?
Probably it would at least make your goal difficult, if we assume alkaline batteries. Let's take some numbers from an Energizer application note, which says that the series resistance for a AA might be around 200mΩ. This value will increase as the battery is drained, and also depends on temperature and frequency, so see the application note for all the details.
If we use that 200mΩ number, then with 16 batteries in series, the total series resistance of the batteries is
$$ 200\mathrm m\Omega \cdot 16 = 3.2 \Omega $$
If you were to draw 1A from this, the voltage drop across the battery's internal resistance would be (by Ohm's law):
$$ 1\mathrm A \cdot 3.2\Omega = 3.2 \mathrm V $$
Assuming a nominal 1.5V across each cell, the output voltage won't be 24V as you'd expect, it will be:
$$ 24\mathrm V - 3.2 \mathrm V = 20.8 \mathrm V $$
While this might work for a short while (pretty soon the batteries will drain, and the internal resistance will rise), it's not terribly efficient. The power lost in the batteries, through their internal resistance is:
$$ 3.2 \mathrm V \cdot 1 \mathrm A = 3.2 \mathrm W $$
The power delivered to the load (your motor) is:
$$ 20.8 \mathrm V \cdot 1 \mathrm A = 20.8 \mathrm W $$
The efficiency is thus:
$$ \frac{20.8 \mathrm W}{20.8 \mathrm W + 3.2 \mathrm W} \approx 87\% $$
And again remember, these are for fresh batteries. It gets far worse as the batteries discharge, even before they are dead.
While 87% efficiency might work, it won't work well. You are making a thing designed to fly. Flight is a lot of work, and an inefficient system means you have to carry more battery weight. The problem is compounding, because more weight means you also need more thrust, which requires more energy, thus more weight. While you can in theory make it fly, it might end up being pretty huge.
Would it be better to hook up three 9v batteries in series?
Probably not. 9V batteries achieve their high voltage in a small package by containing internally several cells in series. Knowing that an alkaline cell (any kind, AA, A, C, D...) has a voltage of about 1.5V, we can infer that a 9V battery is 9V/1.5V = 6 alkaline cells in series. However, each of these cells is much smaller than a AA, and each will have a higher internal resistance.
In order to decrease the internal resistance, you could do a couple things. The first would be to wire batteries in parallel. Two AA cells in parallel have the same voltage, but twice the stored energy, and half the effective resistance as one AA cell. So, you could make 16 pairs of parallel AA cells, then wire those 16 pairs in series. However, this arrangement also has twice the weight, which is really not good in your situation (flight).
The better solution would be to use a different battery chemistry. Alkaline batteries are good because they are cheap, and that's about it. In terms of internal resistance, they are poor. They also have not great energy density, meaning per unit of weight, they contain less energy than you could have for the same weight of some other chemistry of battery. Again, low energy density is really bad for your situation.
Among the other chemistries you might consider are:
With lithium polymer batteries and chargers being widely available from RC hobby suppliers, and their superb energy density, I'd recommend them first for your situation.
