Kind of a hopeful title, above -- I know this'll work for Anthurium, but I'm having to take the word of the internet for the Schlumbergera parts. But it'll probably work, and even if it doesn't, I'll give you an excuse to eat cupcakes, so really you'd be a fool not to try, right?
So. First you need to get some fruits, to take the seeds from. With Schlumbergera, that's relatively straightforward -- get two plants of different varieties blooming at once, transfer the pollen (usually white or yellow, on multiple stamens that stick out of the flower) from one plant to the stigma (there'll only be one of them, which will also be sticking out of the flower; usually the tip is fuchsia) of the other, and vice-versa if you feel like it. This is as easy as touching them together. The difficult part is the waiting for it to mature, which on my plants took something like 6-7 months.1
Anthuriums are tougher. Though the flower2 can stay on the plant for two or three months, the individual flowers aren't necessarily dropping or accepting pollen for a lot of that time, and the pollen, once dropped, doesn't stay viable for long either. So you not only have to have two plants blooming at once, they have to be in the right stage of blooming for this to work.
Anthurium flowers are both male and female, but never at the same time; they're female first, and exude tiny droplets of fluid when ready, which kinda looks like this:
The bumpiness of the spadix is my main cue here, though for reasons I'll get to, I don't worry so much about getting the timing right.3
Pollen-shedding is a lot more obvious. I don't have a photo of it, but it's easily visible; it looks like white or cream-colored dust, and if the flower's sitting so the spathe is below the spadix, pollen will collect in the spathe. Pollen-shedding can go on for several days. (Couldn't find a reference for how long the flowers are receptive when female, but I think it's also more than one day.)
Since I have several different varieties blooming at any given moment,4 and since I'm not trying to breed for any particular characteristics, I just run around the house with a small paintbrush, brushing each spadix in a more or less random order, making sure to collect and transfer as much pollen as I can see. This worked so far, possibly even a little too well.
Anthurium fruits take about the same amount of time as Schlumbergera to develop, six or seven months.
When both are ripe, go buy yourself a dozen cupcakes. You don't have to eat them all yourself, but you need to buy several so they'll give you a clear plastic container with twelve cupcake-sized depressions in the bottom. (If they don't sell cupcakes that way, you can use pretty much anything that has a transparent top and a bottom that's at least a couple inches deep.)
When the cupcakes have all been eaten, relocated, or otherwise disposed of, fill the bottom of the tray with a couple inches (~5 cm) of vermiculite and moisten it. You want the vermiculite to hold as much water as it's capable of holding, but not more than that -- you don't want loose water sloshing around in the bottom of the tray.
So. You'll know Schlumbergera fruits are ripe because they'll change color (my pink/red variety formed fuchsia fruits; the yellow variety formed white fruits that eventually turned light pink). They'll twist off the plant without a lot of resistance.
Anthurium fruits begin the same color as the spadix, change to green as they begin to grow, and then are eventually orange. They're fairly easy to get out of the spadix. Some will pop out neatly if you wiggle them with a thumb, others (especially the overripe ones that have shriveled like raisins) might need a gentle tug with a thumb and forefinger. A substantial percentage of them, though, will burst on you if you try to wiggle them or pull them. The insides are sort of sticky/gummy, not juicy like you'd expect, and so you wind up with odd-smelling5 orange gunk all over your hands. If there's a good way around this problem, I haven't found it yet.
Removing the seeds from Schlumbergera is simple. You cut the fruit open --
-- then scoop out the seeds. I washed mine off in a glass of water first, to get as much of the pulp off of them as I could (it can't help anything, and it might cause fungal problems later), which means I can show you how tiny they are.
Getting them back out of the glass of water was tricky, by the way, but at least they had the courtesy to float.6
Anthurium fruits each contain only one or two seeds, embedded in the orange gunk, and the orange gunk sticks to them terribly, so I had to wash them off in the cup of water again. They sink, which is convenient: you can stir the water hard and knock off the pulp, then pour the water and pulp into the sink, then fill the glass again and stir. Repeat until you feel like you've gotten enough pulp off, then pour the seeds out onto a paper towel.
Both Schlumbergera and Anthurium seeds need to be in close contact with the vermiculite, so they'll stay moist, but not necessarily buried by it. The Schlumbergera seeds are small enough that I just fished them out of the water with a plastic rod and touched them to the vermiculite until they stuck; with the Anthurium seeds, you can move them around with your fingers. I did press some of the Anthurium seeds into the vermiculite lightly with the plastic rod.
After that, you put the cover back on, set the container in a suitable spot (bright indirect light or filtered sun, between 75-85F/24-29C) and wait. You should see some growth within about a week on the Anthuriums;7 I haven't tried Schlumbergera before, so I'm not sure how long those take, but everybody on-line says they germinate fast and then don't do anything for a while.
The Anthuriums can sit in the container until they start to touch the lid, at which point I recommend removing them gently and potting them up in a decent potting soil, ideally one with some coarse, unchopped sphagnum moss mixed in (about two parts potting mix to one part sphagnum). You may wish to pot several seedlings together, as individual plants tend not to be very full-looking and this will hedge your bets in case some of the seedlings die later. I read on-line that it usually takes 3-4 years for a seed-grown Anthurium to flower, and indoor-grown plants may be a bit stunted from being inside, so it could take even longer, making this a long-term project. On the other hand, when they do bloom, you should have a variety of interesting colors, sizes, and shapes to look at, particularly if you take the sort of anything-goes approach I do with my own crosses.
Schlumbergera seedlings in good conditions should be large enough to take cuttings from in about a year; they'll also take 3-5 years to reach flowering size, though cuttings may flower sooner.
None of this is especially practical stuff -- for the time and effort invested, you'd be better off going to the store and buying a couple dozen plants yourself (especially with Schlumbergera, which tend to be cheaper than Anthurium). I'm bothering to try because both plants do well for me, and I like starting stuff from seed, and I'm intrigued by the possibility of seeing something new and different. Even if I'm probably setting myself up for a disappointment, it's a disappointment that won't hit until at least 2014, by which time we'll be so deep in the zombie apocalypse that I won't even care. Until then, I get the hope of seeing something really cool in the future. Seems like a fair payoff either way.
1 My understanding is that fruit development depends on the temperature. Fruits in warmer spots ripen faster than those in cooler spots. I should possibly also note that just in case the seeds from this attempt all succumb to mold or something, I've only started the seeds from one fruit, and have three more left if I need them.
2 Technically an "inflorescence," not a "flower:" Anthurium flowers are made of an insect-attracting part (the colored bit, called the spathe, which is actually a modified leaf) and the many tiny flowers, which are packed tightly together in a column (the spadix). I've never tried to count the number of flowers on an Anthurium's spadix, nor do I recommend that anybody else try ('cause really, don't you have better things to do with your time?) but my estimate is that a couple hundred is probably pretty common.
3 It's probably also important to mention that a number of websites I looked at while writing this emphasized that fluid production usually takes place in the early morning. I don't function particularly well in the early morning, but I've gotten Anthurium seedlings anyway, so I don't think this is probably that critical of a detail, but I read it, so I'm passing it on.
4 At the moment, I have thirteen varieties, seven of which ('Gemini,' NOID pink, 'Orange Hot,' 'Pacora,' 'Pandola,' 'Florida,' NOID purple) have flowers. I also have a flower on A. crystallinum 'Mehani,' but I don't expect to be able to hybridize 'Mehani' with any of the others: I think it's not closely related enough to the cultivars I have (which are mostly hybrids of A. andraeanum, A. amnicola, A. lindenianum, A. antioquiense, and A. scherzerianum, in varying combinations) to be able to cross. Though I'm trying anyway.
5 My closest point of comparison on the smell is that it smells like corn pollen: sort of an innocuous, barely-there smell that's pleasant without having any real qualities you could pin down.
6 Sidenote of interest: I tried tasting the fruit once I'd gotten the seeds out, and it tasted like something once-interesting that had been watered down to the point of unrecognizability. I didn't actually chew any, just touched my tongue to it, so there could be a more pronounced taste if you chew them. Maybe I'll try chewing when I do the next three fruits.
7 In point of fact, some of the Anthurium seeds may have the beginnings of shoots when you take them out of the fruit. This is normal, and just means those are more eager than the others.