Boost your Social and Connections skills to level 5 (minimum). This will give you a great start towards standings, and help increase them faster as you do your missions.
Do all the "Career" agents missions (press F12 and select the bottom option), and do as much of the Data Centre agents as you can (basically collecting or buying pirate tags) for both your own faction and the allied faction.
Get the ~40m ISK Research Project Management science skill, train to lvl 4 (or lvl 5 if you have time to spare) and get several R&D agents (to access the agent's research abilities you need a paricular science skill as well). Make sure you do a mission for every one of them as often as possible (maximum of 1 mission every 20 hours per R&D agent), and cash in your R&D points for datacores before you change up to a better R&D agent. The missions still count towards the 16 missions required for a storyline mission (i.e. faction standing boost), and the R&D missions are always "give me tritanium" or "deliver this package up to 4 jumps away". (The storyline missions after 16 R&D missions are courier or trade too.)
Find a system with multiple courier, mining and trade mission giving agents. If you have to decline one mission then use another agent for the next 4 hours. eve-agents.com (provided by Chribba) is easily the best site to go to if you want to find agents of a certain disposition.
On top of that, make sure you can fly a combat cruiser (at least) and do the lvl 1, lvl 2 and maybe lvl 3 COSMOS missions for your faction and your allied faction.
(Just FYI, Gallente and Minmatar are allies, and Caldari and Amarr are allies. At the moment.)