New to vinyl - How to get the best out of old classical discs?

Definitely not attached to the Ortofon (or any other part of the system). Some people seem to like Goldring and Denon carts for this type of music, though on paper they look very different.

Genre of music is irrelevant for your cartridge, a good cartridge will play all genres of music equally.

I think a better cartridge and a better phono stage would be the first step.  Perhaps you could find a used phono stage in order to stay in budget.  Something like the EAR 834 would be nice.  

A set of $30 Nobsound springs under that table will do wonders for all your music but especially classical where ringing imparts a hardness on the strings. Your turntable is light enough to only need about 3 of the 7 springs per each pod. This will leave you with enough to make another set to use under your phono stage or amp. Either one will benefit, simply try under each to find which you like the best. $30 to improve two components makes this the bargain of the century. 

Spend the rest on as good a phono stage as you can find and afford. A good EAR will indeed be nice. Don't be afraid to spend, a good phono stage is huge. Especially since then with a good one you will be in a much better position to appreciate the benefit of a better turntable when you are ready for that later on.  

Alternatively you could do the table now and phono stage later. There really is no right or wrong. Never talk a man out of a better turntable. A lot of it has to do with what you are able to find. But seriously, no matter what: Nobsound.

Hahaha. I can’t believe I’m reading about springs again, it’s almost in every thread already. Unbelievable.

Nothing will change the sound more than a good cartridge in your current set up @bluemountain.