Should speakers be upgraded before preamp and amplifiers?

Maybe look at integrated amps from McIntosh.  You will want to buy an upgraded amp then align with your speakers.  As an example; if your amp has 200 watts, then buy your speakers that will match at 200 watts maximum.  Other wise you could end up with an amp at 200 watts and your speakers at only 150 amps.

OK, let me ask...why do you want to upgrade?  What do you want to achieve sound-wise?  What type of music are you drawn to?  Format?  Do you have a good dealer nearby where you can demo different pieces of equipment to see what each brings to the equation?.  I think you need to start the journey of identifying what it is that you want to achieve and then begin putting the pieces together.  Speakers bring a lot of character to the overall presentation, but then the source has a lot to do with it as the speakers present what is presented to it, even the interconnects and cables can have a lot to do with it.  So, having a good idea of what it is that is important to you sonically is a good starting point.  You cant figure how to get there unless you have a good idea of where 'there" is.

First step is dump that 50+ year old turntable and cartridge!

everything matching well together is key...though sometimes the buying process can be thrown off by a super good deal ...