I've had good luck with small plants in a peat and APS mix. I haven't done this with newly sprouted seeds (I just can't seem to grow from seed) but I've used it with newly deflasked TC plants. Milled Sphagnum is pretty much nothing but trouble in my experience; it's like an algae/carpet moss magnet.
The most reliable method I've found for keeping plant pests away is thoroughly rinsing the media before planting anything in it. Back when I had a yard with a big shady porch that was otherwise unused, I would make up pots of peat-based mixes in the late summer and leave them outside until the following spring. The perpetual Washington rain would rinse out the soluble nutrients in the peat to the point that only CPs stood a real chance of growing in it. When I brought them inside to put tropical plants in, I would pour boiling water through to kill any hitchhiking invertebrates. Worked pretty well.
When you don't have space outside or the patience to wait six months to pot something, just make sure to rinse your peat thoroughly - two or three times is preferable, from what I gather. Peat moss is relatively low on nutrients compared to potting soil, but out of the bag it still has lots of byproducts of its decomposition from LFS to peat, and possibly mineral-rich silts and such too.
Now that my outdoor space has been reduced to 8' x 11', I don't have room for the above method. So instead, to prepare peat I'll put it in a big stock pot, pour boiling water on it until it's covered, let it soak for a few minutes, then strain the water off. (This tea-like water can be saved and used as a sort of fertilizer for Sarracenia; it's very acidic and contains tannins that help Sarrs produce their red coloration.) Then I'll refill the pot with cool RO water, and after a few minutes, drain that too, after which I'll squeeze out the peat to remove as much water as I can. Then, unless I'm feeling impatient, I'll do one more rinse with cold water.
Alternately, when I'm feeling really impatient, I'll make up my pots first (using wetted but not rinsed peat) and then pour boiling water through after, running cool tap water on the outside of the pots to keep the plastic from heating up too much. I'll let the mix absorb the hot water and steam for ten to fifteen minutes, then rinse again with cold water. This method isn't as effective because the peat doesn't get wrung out between rinses, but I've taken to using low-growers such as Utricularia as ground cover to block out weedy things like algae and carpet moss.
Don't boil your peat moss on the stovetop. I've done this a few times, when making "peat tea" for my Sarracenia, and once tried using the leftover peat moss in pots. It worked like normal at first, but seemed to compact and break down faster than normal. I'm not certain that cooking it was what did it, but it would make sense, and I'm pretty sure that's the only thing I did different. Soaking the peat in boiling-hot water is more than enough to get the nutrients out and leave it mostly sterile.
For pots that are already made up and planted but experiencing carpet moss or algae problems, you can get rid of the pests or at least curb them if you work at it daily. For carpet moss that's well established, it needs to be picked out with tweezers or something. The largest gobs of algae are also best removed this way. Once you're left with sparse little bits of moss, or thin layers of algae, you can kill them with a spraybottle. Use plain water, and set the nozzle to a stream instead of a mist. Spray the moss and algae hard to break it off the surface of the media. If your media comes up to the rim of the pot, sometimes you can just let it run off. But otherwise, simply spray the infested areas until you start to stir up some of the media underneath. The idea is to cover the broken bits of algae and moss in mud, so that it can't get light. It will struggle to grow and use up the nutrients in the media around it, and each time it comes back, there will be less for it to eat. It requires lots of attention, but isn't too hard to do once you get the hang of it, and eventually the plants that you actually want in the pot will be big enough to outcompete the pests (in most cases.)
~Joe