Bang for Buck

Limited budget upgradeWill I get more bang for buck with a Integrated Amp upgrade or a Phono Stage and Cartridge upgrade?Buying new speakers and have some budget left over. This is a single source, Phono only, system.
source > amp > speaker. Upgrade cartridge, then phono stage. Why not look at an integrated with a phono stage?
What are the components in your current system? The ones you are considering upgrading. And what is your allowable budget?
Agreed @reubent . I think the budget is going to dictate this answer.  
Right. Budget. If the budget is $150 then your best bet is probably a Blue Quantum Fuse, set of BDR Cones, maybe a Synergistic Research Master Coupler power cord. Something like that. If it is $3k then your best bet might be the Herron VTPH-2A or Koetsu Black Goldline. Unless you already have a great front end like that.

Which is why reubent wants to know, what is the rest of the system?
I was hoping for a global answer rather than a manufacturer/model recommendation.Eventually all will be upgraded, but for this say around $3000.
Thanks noromance, that’s my logic. I’m helping a novice friend who like most newbies goes for the big shiny first.
With $3k budget, big shiny preference, analog only, this to me is a no-brainer: Herron VTPH 2A phono stage. Sorry that is not a global answer but the Herron is the best component I ever bought, at any price, bar none. Your friend will be stunned. 
Lots of great choices for Integrated including Phono Pre.
A few demos should be convincing.
The global answer is a meticulous setup and correct room treatment will give the biggest bang for the buck by far. The setup and room will determine at least half of what you will hear. No gear swap will yield that great a difference.

Thanks Tomcar,I agree with you completely. The room is currently in dedicated listening room remodel process, the impetus for the gear upgrade.

What are the components of the current system?  Without that knowledge, it's completely impossible to say which element would be the "weakest link" and would be the first to upgrade.
This is an impossible question to answer without testing the potential replacement components in the system. I've seen situations where a system might have an obvious weak link but it turns out the biggest change comes from replacing something completely different.

I just repaired an old Counterpoint SA-3000 which includes a phono section.  It is NOT an integrated unit just a preamp.  The phono was quite good having a really nice, sweet mid-range, So that may be an option plus they are not that expensive and can be upgraded for cheap.

Happy Listening. 

Synergy is an important component, thus no generic solution can be applied to an unknown system.

Low impedance speakers may mitigate fir a better amplifier, are tubes either pre amp or amp a choice, what cartridge and tone arm are in use, etc.

One cannot determine the weak link most easily upgraded in the dark.
Loudspeakers and cartridges being electro-mechanical devices Are the most likely to make significant changes in sound quality. Chaaz just got himself a new set of speakers. Upgrading the Cartridge will have more of a positive effect on the sound than a sideways swap of integrated amps unless he was going from a pea watt amp to a very powerful one.
If you are searching for a global/generic recommendation, then always choose transducers first. The cost to benefit ratio applies more reliably with mechanical/electro-mechanical devices over pure electronics.
It all matters, of course, but bigger chunks should be spent on mechanical devices.
Ahhh...mijostyn beat me to it while I was typing!
Did two threads get merged here?  I seem to remember several of these posts on a completely different thread (including mine).
 The Best Bang for the Buck ? Try the Bunny ranch in Carson city in Nevada !
OP- dedicated room is a great start. Next is room treatments. Then a dedicated line. After that, any gear swap yields typically small but definitely noticeable changes, IMO
Everyone is making all these recommendations without even knowing where he is starting.  

Depending on what you have, and what you want to achieve will make a difference in what you do next.  So, you got new speakers... what are you trying to do next?
Go for the phono stage and cartridge next