Lo and behold, there are roots already growing in! I can't believe how fast impatiens put down roots from cuttings. It's sort of amazing. So, yeah, if you're a total newbie at propagating plants, start with impatiens.
Just cut off some healthy stems, plant them in soil, water carefully so that the leaves and soil are always kept moist for the first 1-2 weeks (so pick a container with good drainage), and you'll have no trouble growing a whole batch of new plants. Only give them indirect sunlight too. The direct sunlight will dry them out too fast, and you want to simulate a moist environment that will keep the cuttings alive and hydrated until they can put down roots to get their own water source. See how I totally soak the leaves with water? I'm not giving them a chance to dry out at all.
The full propagation tutorials with photographs are here.
I checked on the dahlia, petunia, and fuchsia cuttings too, but none of them have roots yet. Impatiens are just Speedy Gonzales!