Qualifying Statement: Unfortunately there doesn't seem to be any way a new player can get a Sol permit and still be considered a "new" player; there is too much reputation required to earn the permit for a player to be successful in any amount of time that would be considered short. It's a grind no matter what, but the good news is that you can be efficient about it.
As long as you have a big bank account, I think you're looking for the Donation Mission strategy, which will cost a lot of credits but require the least amount of travel, confrontation, and waiting. Finding systems with multiple stations containing Federation-friendly minor factions should provide you with a handful of Donation missions every time the bulletin board updates. They just require you to accept them, then turn them in for a fee. It's the fastest, safest way to rank up, if you have the money, and the Donation missions don't seem to be any more or less plentiful than the other types. More details below on how much it's going to cost you (spoiler: kind of a lot).
In terms of a quick and easy way to rank up with the Feds, there really isn't a quick way to get a Sol permit, and the definition of easy seems to vary depending on your play type. After doing some research, I've listed the known methods of ranking up with the Federation, listed with their probably time / safety / cost factors. I've gotten my information from the E:D wiki and a few forum posts of varying age and legitimacy, but as you will see, there is no clear-cut answer...
The General Gist
To gain rank with the Federation, you'll want to be in Federation space, and you'll need to find stations that contain missions for Federation-friendly minor factions (expand the faction list to see all of them, though I can't say how accurate/complete this list is with how dynamic the universe is). Complete minor-faction missions to gain reputation, and at certain ranks you will gain faction-specific privileges. For the Federation, a Sol permit is achieved at Federation Rank 4, Petty Officer, but not all ranks give you permits or privileges (some give neither).
Warning: Pay attention to who you're working for. Doing missions for an opposing faction lowers your rep with your intended faction, so be clear about who your allies are before accepting / completing missions.
The elephant in the room: Server-Scumming
I think I just invented that term, but lots of posts about grinding faction ranks are griping about how long it takes, and they all seem to have resorted to the same strategy in one way or another: finding a place that has missions they like, accepting them, switching between solo and open play to spawn new missions, then accepting those as well. I haven't tried this, but I wouldn't judge you if you did, and it sounds like an efficient way to eliminate the wait for missions to re-spawn (or at least do more at one time). Here is the step-by-step, or the best summary I could find:
When you have no more missions you can take from the bulletin board, simply exit to main menu and switch to solo/open play. Go to the bulletin board again and new donation missions will show up. This works in 90% of the cases but not always. In my experience, once or twice per hour this "relog" tactic doesn't work so you just take a brake for 5 minutes and come back. They should have refreshed in 5 or less mins.
I can't vouch for the time limit mentioned, and I'm not in a good location to test it, but this strategy could very well give you a boost in the time-efficiency department.
Types of Missions
I can't find a comprehensive list of all Naval Rank mission types, but rank grinders seem to favor the following, depending on their play style:
- Trade: Retrieve / Transport / Deliver goods
- Attack: Ground base assaults
- Moneybags: Donations
Depending on your ship and equipment, Trade vs. Attack can be self-explanatory: You need a cargo hold for Trade, and you need an SRV (as well as Horizons) to assault a ground base. That being said, the Trade missions usually require a very modest amount of cargo to be moved, so an attack ship could cheaply outfit itself for a mission without wasting many credits. The Attack option just requires an SRV and the gumption to go looking for a fight, but that's dangerous and possibly expensive if it doesn't go well. The Moneybags strategy is obviously the most expensive option, but the safest and possibly fastest method available.
- Speed: Slower
- Danger: Medium
- Cost: Medium
These missions will most likely ask you to deliver goods to a nearby location, or more often deliver goods to your current station that aren't available at your current station. These are usually purchasable-only commodities, like weapons or food (if you're lucky, food is sometimes available at the stations that request it, but not often). Most pilots say that 20-30 cargo slots will provide enough storage to complete the high level missions you get around Rank 4, but the lower missions may ask for as low as 3 or 4 units of cargo. Try to troll around a system for various missions and you may find that the goods you need to deliver to one station are available at another, and that second station is asking for goods available at the first station. More often, you may need to go to a nearby system, so finding two populated systems nearby each other, with stations in close proximity to the arrival point, can expedite this significantly. Additionally, server-scumming may net you multiple missions for the same station or resource.
- Speed: Medium
- Danger: High
- Cost: Varies
Some federation missions will request that you attack a nearby surface base run by an opposing faction. Many players consider these plentiful (I can't say the same, but I don't look for these types of missions either), easy to complete, and they always net you a certain amount of credits depending on the mission level (whereas the Trade and Moneybags options will most likely cost you a medium to high amount of money in the long-run). In one of the posts I found, two users said they thought that these base attack missions gained more rep (allegedly 4X as much) compared to the donation missions, but I couldn't find another source to suggest that.
However, if you are unsuccessful, your SRV is destroyed and the replacement cost (5,270 CR) might eat into your bank account after a few misadventures. So, for a successful base attacker, this is a low-cost, high-profit method. If you're not very good at ground combat, it could be a high-cost, no-profit, very slow method of grinding. There's always the chance that server-scumming could get you duplicate missions for the same base, or missions for multiple bases on the same planetoid, but I guess that would depend on number of nearby ground bases held by unfriendly factions.
- Speed: Fast-ish
- Danger: Low
- Cost: High
Everybody needs money, and minor factions are no different. I've seen it suggested in several posts that lower rank donation missions cost, on average, about 10,000 CR. The higher level missions may run upwards of 2,000,000 CR. This is the most expensive method (2,000,000 CR covers quite a few replacement SRV's, or units of food), but being at the right station in the right system could get you multiple donation missions for multiple friendly minor factions (and server-scumming could possibly double the amount of missions). If you can put your money where your mouth is, the ability to pretty much instantly accept and complete several missions at once, all without traveling / fighting / purchasing commodities, would make this the fastest grinding method available.
So, to summarize, the Trade strategy is middle-of-the-road, not too slow, costly, or dangerous, but a little of all three. The Attack strategy is the biggest gamble, but can be the most fun and profitable. The Moneybags strategy is expensive as all get-out, but you can do it with relatively little travel, and little to no confrontation at all.
Full disclosure: A Sol permit isn't really necessary for my personal play-style at the moment, so I haven't pursued one, nor have I tested these methods personally. Feel free to comment if I've missed anything, or if an update has changed the way any of this works.