How do I keep my ship alive long enough to kill the bad guys?

Solution 1:

The Rifter is one of the fastest frigates available, as such you should "speed/signature tank" it, which means you try to outrun enemies rather than taking the full hits. The faster your ship and the smaller your signature radius the less damage you take.

You'll still get hit, but you're able to negate these by either of two ways: active tanking (Shield Booster) or passive tanking (Shield Extender). For new players, I'd consider active tanking to be the easier solution, as such I'd suggest the following fitting:

High Slots

• 3 identical Autocannons; the biggest you can fit (200mm or 125mm)

Mid Slots

• 1MN Afterburner (preferred) or 1 MN Microwarpdrive
• Small Shield Booster
• Invulnerability Field (or a specific Shield Hardener, if you're going to switch it based on enemies)

Low Slots

• Damage Control (significantly improves your survivability for minimal costs)
• 2 of the following (based on personal preferences; speed over damage, etc.): Gyrostabilizer, Nanofiber Internal Structure, or Overdrive Injector System

In case you'd like to try a passive tank, you can find a passive PvE fitting on the UniWiki. You could as well use armor tanking, but I wouldn't recommend it for starting the arc as a new player. While it offers protection, it also lowers your damage (by using up your low slots you could use for damage modifiers like Gyrostabilizers) and makes you slower (so easier to hit).

Fighting

• Try to pick enemy after enemy, do not rush straight into them, as they'll easily swarm and overwhelm you (especially Rogue Drones). These missions are harder than the typical level 1 mission you might have flown before.
• Always keep moving. If you stand still or slow down, your enemies will have a far bigger chance to hit you more significantly, especially when in a frigate.
• Only use your Shield Booster when you have to (e.g. shields drop below 50%). This way you're saving capacitor energy compared to running it all the time.

Later on

Keep in mind that some of the later missions in the story arc get rather tricky, there are three things to note:

• If you see elite Rogue Drones (name starting with "Strain", e.g. "Strain Infector Alvi"), try to take them out first. These drones have Warp Scramblers, that will keep you from fleeing the site in case something goes wrong. This is the first and only encounter of Warp Scramblers for a while (they're rarely used below level 4 missions).
• In one of the last missions you'll have to fight a named Battlecruiser that uses Energy Neutralizers against you (removing your capacitor energy). This might be a tough fight for a new player (in fact, I consider it the hardest enemy of the whole story arc), so consider bringing help or fielding something bigger than a frigate.
• The last mission is rather easy, but you'll need quite some damage to break the boss' tank (his damage is pretty low, so you can use trial & error here).

Solution 2:

What you're referring to is called 'Tanking' in EVE Online and there are three types of tanking you can use.

Shield tanking uses your shield and improves it by adding modules to your midslots (aka medium power slots).

Armour tanking uses your armour and improves it by adding modules to your low slots.

Speed tanking (or dodging as it is know in other MMOs) improves your ships speed and agility via mid and low slots and is combined with orbiting your target so fast that their weapons can't track and hit you. This is the best choice for a Rifter.

Once you know the basics it's just a matter of experimenting with modules and there's a comprehensive guide on the forums which will fill you in on the details.

To fit a ship for speed tanking in missions you basically need an Afterburner module in a midslot, two Nanofiber Internal Structure modules in low slots. In addition adding Overdrives or Inertia Stabilisers can help.