The AKG K240 is a great introductory headphone. Out of curiosity, though, are you using them for casual listening, or music production?
On a side note, I've had my K240 Studio (Austria-made versions, which I understand to be 100% identical to China-made versions) for something like 7 years now. I've replaced nothing on them and that's with cataloging thousands of hours using them for music production. Only issue is the ear pads are hardening and cracking from usage. Treat them gently, though. And get a headphone amp because, at something like 60 ohms, they need to be driven well to get the most out of what they can offer.
My own opinion is they're excellent for an exact representation of the music. What you hear is what you get: no sugar coat. That's the job of studio headphones. These headphones sound awesome accurate. Easily ranked in the top tier of best studio headphones under $200, and I wholeheartedly believe they can give a couple $300+ headphones a run for their money. The flat response they provide is something I prefer from my headphones. I have a car with an aftermarket audio system as well as a Canton home audio system when I want my music to be engaging.
Unfortunately, I don't find them comfortable because my ears get pressed in by the drivers and I have to take them off after about 30 minutes. That's just my personal opinion. But, lots of people like the comfort. Lots of people dislike Bose, but I honestly like Bose (mediocre sound..i'll admit...but their comfort is unmatched). So, don't let my opinion on their comfort dismay you.
Which leads me to say this: headphones are largely "your mileage may vary." However, studio headphones are a perfect headphone type to start with, whether as a casual listener or musician. A flat response lets you know what you want from a sound signature (bass boost vs flat range vs sharper mids/highs, etc.) as well as from the actual headphone itself (comfort, closed/open back, pressure of the headphones on your head, etc.). From these headphones, you can start looking in the right direction. But, spend a good amount of time with these headphones. Don't try to CREATE an opinion about them. Let your opinion form naturally without actively trying to. Just play your favorite songs/records and let the music take you away!