I have read the Eve Uni wiki articles, but I am still a bit confused.
How could I get really high standings with the 4 major factions (Amarr empire, Caldari State, Gallente Federation, Minmatar Republic)?
Should I run distribution missions? (I tried it, it didn’t seem to work).
Should I run security missions and decline the missions which are against the other major factions (I tried that and all of my storyline missions were against one of the other major factions)?
Should I farm rats for their tags?
My plan right now is to train the Social skill to 5 and do all of the careers missions and the SoE epic arc, but apart from that I don’t see a clear path to increase the standings with all four of them.
In short: epic arc cycling. Each faction has its own plus Sisters of EvE arc where you can choose the faction at the end. Those arcs give a high faction standing boost as reward without a penalty to the opposing faction.
As you can repeat the arcs every 3 month, it will probably take more than a year from scratch. You also need to farm the standings for arc access first, and train the ship skills to run them. It can be a long term goal.
Whether you need it, is another question. There is little use of overall high faction standing, except with Caldari (Jita fees). I personally have +5 with all four, because I want to be able to use lvl4 locator agents across empire space.
EDIT: Context, having high faction standing with all factions is not foreseen by the game, the regular plot is that you decide for either side Caldari/Amarr or Gallente/Minmatar. That there is a way is expression of emergent gameplay which was accepted by CCP, otherwise they would have patched it out, but they certainly don’t support it.
Run missions, if you’re doing combat missions before accepting the mission. Check if there’s any faction involved in the mission. Many will have the icon for Empire or pirate Faction. If there’s any of those who’s standing you’re trying to improve on decline but only once every 4 hours for each agent. This is due to the fact that the mission involves killing the ships in that faction and lowering standing with the faction. If you decline more than once per set period with same agent. Standing is also lowered. If the second mission involves a fraction you want to improve. Go to another agent and run missions. Every 10 missions completed you receive a message for a fraction mission. These missions can be anything from hauling, mining and combat whiich is random. It’s also these missions that will increase the standings with that faction. There’s also the SOE epic arch missions but depending upon what you choose at the end to improve can also lower standings with their opposing empire faction. Which should only be done if you want to improve standings with that specific Faction. The career agents and theirs 3 agents per faction for each career. Completing Each of the agents missions increases standing. There’s also a social skills that each level trained increases the standings with all factions. At level 4 I believe it’ increases the standings to level 2-3 allowing Access to those missions. Mining and hauling missions don’t involve combat with a Faction and no risk of lowering the standings with other Factions, only combat missions do this. Also you have standings with each Faction corp and each clan or sub faction such as the brutar. These standings go up faster with each mission for the corp/clan that you completed but it’s only the Faction missions after completing a number of missions that will increase the standings with that Faction. You can get more information about this from Eve University Wiki site.
That’s every 16 missions of the same level for the same Faction.
There shouldn’t be any Faction standing loss for completing the Epic Arcs, however some of the missions do incur negative standing hits for ship kill but those are few and the standing hit is small.