If you liked the SE425 - but want better bass and top end, I'd go straight to the DT880 or the HD600.
For the record, I also had the SE425's for quite a while.
All IMO - from actually hearing them .....
DT880 mids aren't quite as forward. The bass is extended, defined, punchy and fast. Highs have quite a bit of sparkle and are very detailed. Soundstage is quite good, pretty accurate, but not excessively wide or deep. Responds really well to tubes, and helps bring the mid-range out a little. Very slightly U shaped.
HD600 in comparison has wonderful mids, but lighter bass than the DT880, not quite as extended, but still pretty well defined. The highs aren't as shimmery/sparkly, and not quite as detailed - but still nice and bright. Again responds really well to tubes. HD600 slightly more relaxed sound than the DT880.
Both are really comfortable cans to wear. The HD600 can be a bit clampy at first - but over time it seems to relax a bit. Both are quite neutral (not overemphasised anywhere in the spectrum).
Both need amping.
On some of the other suggestions ...... DT990 can be a fun can for Rock - remind me quite a bit of the TF10 - but definitely quite U/V shaped. Mids are not their strongpoint and take a back seat. I haven't spent any time with the new HD650 - but found the older ones were a little too bassy for my tastes and didn't have enough top end. Newer HD650's may have solved this. Great cans nevertheless - I just preferred the HD600.
Although the grados are supra-aural, I love their sound. If you want to dip your toes in the water for very little money, order an Alessandro MS1i online - about USD 100 delivered. It's fantastic value! You can change things a lot with modding and they are a great complimentary can if you plan to permanently have a couple of headphones eventually. I've now given mine a full woody along with a few other mods - and I now have 2 perfectly complimentary cans (MS1i and HD600).
If you want to try the DT880 first - give Jaben a call - ask for Josh - and see if he'd be prepared to lend you a demo set of DT880 Pros to try. He was really good with me and I ended up buying them. For the HD600, keep an eye on the FS forum.
If you have any more questions specifically about the DT880, HD600, or MS1i, PM me.