Missions & Complexes

Ayup. The career missions are excellent and you should do all of them. Don't be afraid to lose a ship so early. They are pretty cheap and you really need to get used to the idea that the ships are the ammo, YOU are the weapon.

There are 8 (I think) exploration missions available from the Exploration Career agent. There's also a Business career agent, hence I would assume he's talking about those, not regular missions.

Be sure to take advantage of the drone bay on your Rookie Ship.
Learn Drones to level-2 and Light Drone Operation. (about a 1/2 hour of training) This will allow you to use 2 light combat drones.
You can send these to attack targets while maneuvering your ship out of the range of the target(s). This is a useful tactic when using a fragile ship against multiple targets. Drones are a big part of the Gallente ship design, so applying skills to their operation is a good choice for a Gallente pilot.

Remove the mining laser off your Rookie ship, and fit a second gun. You can acquire a rookie ship at any time to obtain a second Civilian Electron Blaster by using the method described above. Just go to the fitting screen and strip the fittings off, then get in the ship you intend to use and fit both blasters.

These two things will significantly increase your damage output without taking a lot of ISK or time. You can buy the skillbooks at your school station, or any station that is a school. You can buy light drones pretty much anywhere, they are often available at the school at a reasonable price. Don't both jumping 10 hops to save 100 ISK. You will find that you can make more ISK in the time spent doing missions rather than jumping through a bunch of systems to buy at a discount.

Hobgoblin I is the recommended drone.

Become proficient at using Frigates before progressing, and do not skip Destroyers. They use the frigate skills and weapon systems, and are very powerful.

Only step into Cruisers when you mastered Level-2 missions in a Destroyer. You will learn more about mission progression levels over time.

Once you have completed all the tutorial missions, and that means all of them. Because, yes you need to know how to mine, and do things besides shoot stuff. I recommend you seek out the Sister's of Eve agent in Arnon.
http://eve-agents.com/index.dxd?AgentName=Sister%20Alitura

This will start you on a 20+ mission Epic Arc that can be completed by new players, and will send you on some extensive travels. You will need help with some of the missions. Ask in local chat for help. There are usually plenty of Vets hanging around doing this mission chain in the system, as the chain is repeatable every 90 days, and yes the rewards from this chain are good enough that even 10+ year vets like me do it every 90 days.

Oh my derp, completely didn't think about career agents. Was a long time since I had any contact with them. Maybe I shouldn't give advice to newbies :)

ships are just tools for completing a particular task
disposable tools

pve isn't the most exciting thing over the long haul
in eve other people will want to shoot you
it's ok to shoot them first

the most important ship in eve
friendship

if you do enjoy completing security missions for the npc's
you will find friends in the community channel helpmymission

You don't need a second rookie ship to get another gun. All you have to do is repackage the ship and then assemble it again. It assembles with new gun and mining laser. I have a stack of 24k+ guns somewhere from doing this over and over again with 30 rookie ships.

I have done extensive testing with rookie ships see my bio =). You can hide ships from the insurance system by putting them in hanger containers if there is only one station in system and you don't want to go next door.