Elite Dangerous: New Pilot's Guide and Advice: Ranking Up with the Major Power of Your Choosing

How to Rank Up
I was raised in the Federation and have no time for Imperial hounds or Alliance cult hippies. But it is critical to choose a home system that is aligned with the Major Power you prefer. As you run missions for local factions, you are gaining both REPUTATION and RANK with the related Major Power. You can see this progression on the right hand HUD in your cockpit. You can see your overall reputation for that power on the middle part of the STATUS TAB, and the current rank progression on the right. As you gain rep with the Major Power, you are also getting better paying missions from its local factions.

Ranking Up - Quick Steps:

Pick a well-populated (high traffic), stable system in the Federation or Empire
Pick a station in that system that is close to the jump-in point, and make it your 'home' for ranking up (keep returning there to get new missions)
Run missions or passenger missions for the local factions affiliated with the major power (Look for the major power watermark on the local faction)
As you run missions you will see your rank with the major power go up, slowly
Build to at least cordial rep with the local major power faction(s)
At 100% rank, check the Missions board periodically for missions mentioning the "Federal Navy" or "Imperial Navy" (Not the Passenger Lounge - the main Missions Board)
Complete that mission and you rank up automatically

Ranking Up - the Details:

Major Power Watermark
Not all missions or passenger missions provide rank with a major power. You should look
for the major power logo on the mission list. In the following screen shot you can see the Federation watermark on East Galaxy Company, just under the "7/8". That means East Galaxy Company is a Federation faction and provides Federation rank.

Why Rank Up?
Ranking up allows you to buy ships that are specific to that Major Power, as well as unlocking access to star systems that the powers close off from the public. If you unlock Sol, you can fly through the rings of Saturn!

100% Rank - Find a Navy Mission
Beyond 3.0.3 Update: Frontiers says rank up missions have been moved to the "Follow On" missions section. I have seen some posts saying that pilots are not able to find them. Please comment with your own findings. It may take a while for me to hit my next rank up level.

Keep an eye on your rank, and when you see that you have hit 100% rank, then go to the missions board and look for a mission that mentions your Major Power. For the Federation look for missions that mention the Federal Navy. If you are an Imperial hound, once you hit 100% rank, look for missions that reference the Imperial Navy: Such as Imperial Navy Acquisition Contract, or Courier Job for Imperial Navy Intelligence, or something similar.

To be certain you are selecting a rank-related mission, when you look at the mission, there should be a quote under the happy, smiling face of your faction rep referencing the navy affiliation:

When you complete that mission you will then level up your rank automatically. It may take a while for a rank mission to appear in the market. Just keep looking. Flip the boards if you need to. It's okay if you don't find a rank mission right away. You are still progressing on the next rank. You just won't see the progression until you run the related rank mission.

Why Don't I See a Rank Up Mission?

Is your CURRENT REPUTATION with the Superpower you are trying to rank up with POSITIVE? If not, send your local Fed or Imperial recruitment office a dozen Krispy Kreme donuts; otherwise they want nothing to do with you
Is the CONTROLLING FACTION at your station aligned with the Superpower you are trying to rank up with? The local dudes in charge must be Federal or Imperial before you will find missions
Are you at least CORDIAL reputation with the local factions affiliated with your major power? No? There's your problem. Run missions and get better rep.
Are you hanging out in a remote system, not well-populated, and perhaps with a station state that is civil war or lockdown? Get thee to a more stable 'home system' in the populated bubble of the galaxy! (Quince, for example, does not spawn rank up missions)
Be patient. Flip the board (change your game type from solo-to-open or open-to-solo to refresh your game instance and get new missions in the list). Go run some regular missions for an hour or so, then check back. Navy missions will spawn if you have followed all of the advice in this article.

I'm Not Good at Combat...What If I Cannot Complete the Mission?
There are different types of rank up missions that will spawn. If you do not want to try a combat-type mission, don't accept it. Just be patient and wait for a different one to spawn.

The Niu Hsing Federal Rank Up Method
Note: This method comes and goes with the state of the systems. Your mileage may vary.

The details for this fairly fast rank-up method were provided by Commander PassengerRickyJames. (Reddit scholar /u/PassengerRickyJames) Here I quote his original post from Aug 27, 3303:

"For those looking to grind Fed Rank: system Niu Hsing is once again in the state of Boom. This is better than most (if not all) other Fed Rank grind spots due to the convenient location of endpoints near the stars, and two Fed factions being present, who hand out Data Delivery missions like candy.

For those unfamiliar:

Go to Niu Hsing, Gorbatko Terminal. Medium landing pads only. An AspX/T6/Keelback is a good choice here, though a Python with a goodly amount of cargo space is strongly recommended.
Pick up Data Delivery missions from the two Fed factions going to Col 285 Sector YF-M C8-8 Lebedev Installation. This is a planetary installation (so Horizons required). Cargo delivery missions will be available as well (hence the benefit of a freighter), and they pay quite well too.
From Lebedev installation, pick up missions going back to Gorbatko. There are also missions that go to a planetary base in Niu Hsing, but it's quite possible to fill up your mission stack without bothering with that station. This lets you loop back and forth faster.
Bonus - every so often one of the stations starts handing out donation missions like candy."

Power-Specific Ship and Permit Unlocks

This link provides a side-by-side comparison of the stats for the Federal and Imperial ships:

The Landing Pad: Federal and Imperial Ship Comparison

Regarding the Alliance

Pending Alliance changes? It was recently announced that at least one new ship (possibly more) will be coming from a partnership between Lakon Spaceways and the Alliance - the Chieftain. This will be part of the Beyond series of game updates in 3304. This begs the question - what will be required to access these new ships? A rank-up system within the Alliance? Stay tuned.

Please note that the ALLIANCE does not have ranks or special ship unlocks. The Alliance is a coalition of independent systems that agree to protect each other from incursions by other powers. There is a permit system in place at some Alliance systems. Until you unlock a permit for a system, you cannot navigate there. The Alioth system may be the only Alliance system that currently requires a permit, and it can be unlocked by running missions for the Alioth Independents faction in systems close to Alioth. Once you reach allied status with Alioth Independents, look for and complete a mission that references "Making Friends and Going Places" to earn the permit.

Other Permits
This list of permits is not complete. This is a big galaxy, and I'm just one commander. Commander Swift Arrow maintains an extensive and well-organized list of available permits by type (power, planetary/moon, independents). This link to Commander Swift opens a different Galnet site - a group operating in a heavily populated part of the bubble:

Commander Swift Arrow's List of Permits

Federal Ships
Federal Dropship: