I’ll respond to the edit first. The only reason to join a corp in highsec is for tax reasons. I’ve found many of them to be quite unhelpful and toxic, full of entitled carebears who resent the tiny amount of risk smart players have to contend with in EvE. You’ll enjoy EvE more if you don’t let them pollute your mindset with their cancer, so stay far away from them.
Now onto how to mission.
Step 0: Train Social to 5, then Security Connections and Connections to at least 4. This will greatly accelerate your journey to Level 4 missions. Social and Security Connections ups the standing reward while Connections up the base standings that the rewards add to.
Step 1: Pick a faction, which you did in Caldari.
Step 2: Use the Agent Locator to find a cluster of the highest level agents you can access for your faction’s Navy, that’d be Caldari Navy for you. The navies seem to have the best LP stores outside the FW corporation.
Step 3: Grind. Accept every mission from every agent you have access to in the system containing the agent cluster. Level 1 through Level 3, take them all unless your ship cannot enter the mission. A couple lvl1 missions take just as long as a level 2 and give almost as much standing reward with Social 5.
Step 4: I need to emphasize this, so forgive the all caps. DO THE DAMN STORYLINE MISSIONS!!! Each one gives you a significant standing boost with the core faction, and they come every 16 missions of the same level.
That’s all you really need to be in lvl4s within a few days. Now for a couple tricks I didn’t include.
If you have skills for Minmatar and T2 medium projectiles, the Cynabal will shred up to lvl3s. It’s arguably the best ship for grinding lower level Caldari and Minmatar because the warp speed bonus makes the travel time that much shorter, and that adds up after half a dozen missions. Combined with the instant damage application compared to missiles, and it’s good for blitzing.
Another reason for finding the agent clusters when grinding is so you can keep going even after declining a mission. I’ve found, however, that if you’re blitzing missions like I described above, the standing hit from declining a mission within the timer isn’t noticeable, provided you’re not declining more than 1 per cycle.