How should I upgrade? A new preamp with a phono stage or just a phono preamp?

There are several ways of looking at this and it all depends on what you like and where you see yourself going with the hobby.  Any good stand alone phono stage from any good company is going to outperform the built in phono stage of any integrated/preamp or receiver.  So if you buy a good stand alone phono stage now it would serve you well going forward no matter what upgrades you might consider in the future.  On the other hand, if you simply don't want the clutter, like the simplicity and elegance of a single piece, and there is nothing wrong with that, then go that way.

I had a vintage nikko nr 719 receiver with a built-in phono stage and a technics turntable and it played marvelously. Used with epicure linear speakers. Probably sounded better than most of the stuff out there in the modern age. Components were really built well back then.

New, Phono anything, I recommend finding sources that allow returns.

Have/try MC, use alternate cartridges?

No matter external or internal, I chose to get a separate SUT with some variations of impedance and 3 selectable inputs, thus only 1 cable sut to phono

Entre 100

https://www.ebay.com/itm/134511637279?mkevt=1&mkcid=1&mkrid=711-53200-19255-0&campid=5338381866&toolid=10001&customid=b1b41ac6-d8b4-11ed-9474-333138643664

I chose this Fidelity Research FRT-4

https://www.ebay.com/itm/155407979749?mkevt=1&mkcid=1&mkrid=711-53200-19255-0&campid=5338381866&toolid=10001&customid=3808290a-d8b5-11ed-9a31-326433663330

There are ones with dip switches, internal resistors ,,, they are more expensive generally, I’ve never owned/messed with one of those. My friend’s Melos Preamp had adjustable internals, I changed them for him until we got a preferred sound.

I try to avoid 'better' going for 'preferred' instead.