Should speakers be upgraded before preamp and amplifiers?

Speakers first. You want to mate them with complementary components. While given whatever speakers you choose it is most important to mate with appropriate amplification... the preamp can maters. For instance if you get hyper detailed speakers... you don’t want a lean hyper detailed preamp... mating is very important.

Tube preamp, definitely. This is the component that is really largely influential on the musicality and naturalness... get this one wrong and everything will not sound right. So choose a good tube preamp.

I'm a believer in building one's system around the speakers. 

Absolutely speakers first. Find some you really like the sound of at resellers and friends.  Notice what they’re paired with at careful listening can shape your component decisions also. One of the things to recall about speakers is that bookshellf speakers with stands take up about as much room as floor standers. Bass reproduction desires could steer you towards floor standers too.

@infinid00 Just sent your link to four people..So thanks. In the words of someone. "It's the room!" 

That’s quite a time capsule you have. I had the same receiver with a PL-19 with JBL 100’s in 1978. That being said, ans mentioned above find the speakers you like and then choose an amp that compliments them. At your price point I would consider a new Technics TT and a Bluesound streamer. If you buy an integrated amp and speakers used you would be able to assemble a better sounding system. Later you could get a separate phono stage and upgrade to a better DAC. You can spin your vinyl and stream internet radio without buying a music subscription. But you need to visit stores or a trade show to find what you like.  Happy hunting , Mike.