EVE New Citizens Q&A

Train Connections / Social skills to 4 to get an initial standings boost, then do missions for the highest level agent you have access to. I would recommend refusing Important/Storyline missions as those affect your Empire standings, and just doing the regular missions. Switch agents as soon as you gain access to a higher level agent.

Try to do missions as fast as possible, just complete the objective, and ignore any escorts or extra NPCs - kill the main objective target and get out. If you can manage to buy the hand-in item (f.ex. Tourists, Confiscated Viral Agent, whatever minerals, ore, whatever the mission requires), amass a quantity of those things so you can instantly complete the mission without actually doing it.

Alternately (don't know how fast this is), if you have a buddy that has access to level 4 agents for that corporation, you can ask him to take missions, help him finish them, and have him hand in the mission for both of you. You should get some corp standings from that. It's probably faster to solo your own agents though, esp. since you can start with level 2's and progress to level 3's pretty fast.

Ok do you want to get standings with a corporation such as Caldari Navy or Aliastra? Or to get standings with an empire such as Caldari State or Amarr Empire?

You need 5.0 base (before Connections skill is applied) with an EMPIRE to be able to anchor POSes in high sec. You need standings with just a corporation in order to access its agents and do missions / get LP rewards.

There are hand-ins for empire standings but not for corporation standings (they'll cost you billions to get a decent increase). If you only need corporation standings it's faster to grind missions for just that corporation, rather than trying to raise your whole emprire standings.

Each level trained in Connections skill gives you an instant +0.4; train it to 4 and you'll get approx +1.5 standings, which should unlock most level 2 agents.

From what I know of mining missions, they require funny ore to be mined, but you may still be able to buy it off the market. There's a lot of crap on the market. If you have plenty of ISK, buy the funny ore and pretend you've mined it. Lets you blitz the missions.

If you don't have it already, train Social to at least L3, with L4 highly recommended ASAP, and even L5 if you feel like it.

Go to this page: http://www.eve-wiki.net/index.php?title=Tutorial_Agents
You will want to run all those tutorial agents (15 for each race, 5 per school) AND all those career agents (3 sets of 5 for each race, one set per school).
Most of those can easily be run with ships and skills you can get in a few days after character creation, if you're older than that they should be a complete cakewalk. They should also not take long at all to run, even with barebones skills, let alone with halfway decent ones.

Select the faction//race//empire "allied" with the one you wish to get the standings (i.e. Caldari with Amarr, or Gallente with Minmatar) and do THOSE first.
When you're done with them, move to the target empire and do those too.
This should leave you with well above +5 base, possibly even close to +6 base standings with the target FACTION (and possibly slightly less great but still good for their allies).
This means you should be able to access ALL AGENTS from all corps in the target faction (and also get some pretty nice discounts on broker fees). It's also good in case you later decide you wish to switch agent types, you don't need to grind something else again.

While you do that, get Connections to at least L3, preferably L4, just in case you get bored before you're done with them all and settle for just a +5 effective standing (in which case, you may want to skip the friendly empire and just go for the target empire directly). Also nice if you want to just kickstart to L3 agents of one particular corp even sooner.

P.S. Yes, I realize that's a "plan" to grinding faction standings, which are supposedly harder to get than corporate standings. But only supposedly. This is faster and better, in my opinion. Also, kind of boring (max of 30 of one type of 2-part mission and 6 of other 10-part mission types across two allied empires, all fairly banal and mostly short missions, so you'll probably spend more time moving from agent to agent than doing the missions themselves), but, eh, worth it.
The only drawback to this is that you can only do it once per character (if you screw up your standings for some reason later on, you can't redo those missions to repair it).

If you mean you're trying to get your entire corporation's standings to that point... you probably won't be able to, not by yourself.

Corporation standings are the average standings of the players in the corporation that have:
1. Logged in within a certain time (I think the last month)
2. Been in the corp a certain amount of time (I think a week or two-- this is why Estel Arador jump clone corp kicks you out after a week)

It also depends on _actual_ standings of members, not the ones you use after modification by diplomacy, connections, etc. Though I think corp standings are improved per player based on skills.

So if you grind a standing to 5 and two or more other active players have that standing at 0, you can't get the corp standing to 5.

Bummer.
That's somewhat unpleasant, but not really THAT much of a bad news
The standings gains from the tutorial agents were much smaller than the ones from the career agents anyway, and all of them combined were "worth" less than a single set of career agents. You might not get to +5/+6, but a +4/+5 should still be quite doable, I assume.
On the plus side, that means you have that much less systems to visit in total.