First off, I can't believe anyone listens to those Apple earbuds at all. They're horrible sounding and physically hurt my ears when they're in. Just awful.
Anyway, I've got Sennheiser CX500Bs that are MUCH better and more comfortable than the Apple garbage. I've had them for years and they're still going strong, and they also incorporate a volume control on the cable which sometimes comes in handy. If you can find them (not sure they're still making them) you can probably get them for about $40, but make sure you get them from an authorized dealer because there are fakes out there that are tough to spot. They also offer the CX300B that looks very similar but lacks the volume control and can be had in the low $30. Not sure of how they sound vs. the CX500Bs though, but I'm sure they still sound a helluva lot better than the Apple buds. They both also come with a few different sized ear adaptors so you'll get a good, comfortable fit that will also minimize any sound leaking out that others could hear. I'd categorize the sound of the CX500Bs as nicely detailed and a bit on the warm side due to warm, punchy bass and rich midbass frequencies. These might be a good choice if you listen to a lot of music that's not well recorded or thin/bright sounding.
Probably some of the best 'phones in the sub-$100 price range are the NuForce headphones (see their website). This is a high-end audio company that makes quality stuff. I got their NE700M 'phones as a gift for a friend and he likes them a lot. The buds are actually made from machined aluminum rather than plastic, so they'll probably hold up better than most. Probably not as warm sounding as the Sennheisers but probably more neutral and detailed, although I've not heard them. They're $69, but they also offer models at $65, $49, and $25. Can't see you going wrong with these, but unlike the Sennheiser's they probably won't do much if anything to help bad recordings sound more listenable but good recordings will probably sound even better.
As mentioned above, read up on what else is out there but these are at least two options I have first-hand experience with. Best of luck and congrats on getting rid of those horrible Apple 'phones.
