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Brass: the best kind, but is the most expensive of the bunch. It's very resistant to mother nature, it's soft so it's low stress on your guns, and it is very resistant to deformation and expansion. Not to mention is the only type that is reloadable.
Aluminum. Nice and cheap, but not all ranges allow aluminum because it can't be reloaded. I haven't had any issues with the 200 rounds of 45 I accidentally bought one time. I would never buy it again, though. I can't imagine it holding up better than brass.
Steel: um... some people hate it, some people don't mind it. I'm in the latter part. Normally found on rifle ammunition (whereas I typically see aluminum on handgun ammo), there are reports on the interwebs of people's rifles getting indigestion with that ammo. I personally use it exclusively for my AK. I've had zero problems, zero malfunctions, and plenty of fun. Just like aluminum, it's not reloadable, so some ranges don't allow it.

I can't imagine the striker or firing pin to have any issues with any of the ammo. The primers are all pretty much the same material on the strike face. The extractor/ejector may wear prematurely with steel, but those are wear items anyway. The only issue would be corrosion from the berdan primers. I'm a little leery of the corrosion-free claims, but I've never had a single problem with thousands of rounds through my AK. So I guess the corrosion-free claims really do hold up.