Phono Stage, Cartridge, DeGritter...which one first?

If you were considering any of these changes, which one would make the most immediate improvement without narrowing it down to a specific phono stage or cartridge...interested in some feedback from the analog folks here.
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A cheap ultra sonic solution as suggested by @neonnight coupled with a Project VC-S2 ALU to vaccum the record dry from a distilled water rinse. The Project is $699.

I like to use Tergikleen. 10-20 drops per gallon of distilled water. Then a distilled rinse.

Tergikleen is what the Smithsonian uses to clean and preserve its vinyl records, from what I’ve read.

You’re all in for around $1200.
Listen to some nice clean records to help you decide if a cartridge or a phonostage is your next right move.
The Library oh Congress, the BBC, many dealers and others use the Keith Monks RCM. Dealers use them as a service for clients and some charge. It is a vacuum system that uses a German medical grade vacuum pump attached to a tone arm like arm with a nozzle at the end. Their fluid is also special. Their model without the fluid pump allows one to just apply the fluid with a squeeze bottle. It cost much less. You apply the fluid and then spin the record while holding a brush down. The the arm placed on the label moves like a tone arm in reverse over the record vacuuming up every bit of fluid and dirt. It does this unlike other vacuum VPI types and ultra sonics.
Upgrading to the P8 is a good idea. It is very much like the P10 for a lot less. And improve you phono stage, then think about a RCM.
Clean records are a must.  I have been using a Nitty Gritty for many years and it did a fine job of cleaning the record, visually.  I have focused on my turntable, phono cartridge, and phono stage first and have made significant sonic improvements every step of the way.  Then, I upgraded my record cleaning system to a Degritter.  I still use the Nitty Gritty for used records and once cleaned by that vacuum RCM the record goes to the Degritter.  The results with all of my records (current collection or used purchases) after US cleaning in the Degritter is that I am a little closer to the music.  For example, the music is more detailed in a manner that I am hearing a sizzle cymbal trailing off for a longer period than I had heard it in the past (same record, very familiar with it).  Vocals (again, same records that I am familiar with), if recorded well, I will often hear lips smacking or vocal chords vibrating that I had never before heard.  Many other examples, but that should give you a good idea.  These finer details in the recording, however, will not be revealed by US cleaning if the turntable, cartridge, and phono stage are not up to the task in the first place.
So, as I did, I can recommend that you first focus on upgrading the analog front-end, and then upgrade to a US cleaner.  Until then, be sure to keep your records clean.  The Record Doctor is a very good entry level vacuum record cleaner, low cost, it does a great job, and it's made by Nitty Gritty for them (a quality product).  I can wholeheartedly say that the Degritter is an excellent choice, providing it is within your budget.  It is engineered from the ground up to clean vinyl records.  It is not an off-the-shelf US tank made for cleaning jewelry or other items.  It does the job better for vinyl than any of its peers, but it's also a lot more expensive.  
I hope this helps you make a decision(s).
Clean the records! Be sure your cartridge is aligned properly in the arm and that the bearings are super free. Check the anti-skate so that the cantilever stays centered when the record is playing. I had a friend whose alignment was so bad he thought particular record was a trio, but once we got the cart in alignment, he discovered it was an orchestra!
do you have spacers under the P6 tonearm to level and optimize the VTA of the Hana cartridge?