What's good for classical music is potentially good for Mahler, but what's not definitely won't be good for Mahler, which means Grados will probably not serve most people's taste for classical music here.
RA-1 is designed by Grado to drive Grado headphones. So it doesn't really work that well with high ohms headphones like HD650. Some audio dealers treats the RA-1 as a generic amp, which IMHO is a big mistake. RA-1 is a very, very good amp, but it needs to be paired with the right cans.
I have recently upgraded my amp from a TBH to Perreaux SXH-2. The SXH-2 is simply an amazing amp, and I wouldn't hesitate to recommend it. My options for the upgrade were Rega Ear and the Perreaux. And it was a no-brainer, even tho the Perreaux was more than twice the price. Rega Ear is also very, very good. But overall it actually sounds quite similar to my TBH, so it wouldn't be much of an upgrade.
ucbEE, I see that you have a Indigo for your computer setup. I have never listened to it, but the general opinion is that it's quite good. On the other hand, I doubt it has enough power to drive headphones like HD6XX, not to mention K501. So if you plan to get a pair of those, you might need to get a more powerful amp.
As for cables, I'd say leave that till you've got everything else. You're much better off spending the money on amp, source, etc, first before getting a (usually) very expensive after market cable.