How does missons work today and what ships are best?

Learn how to go each individual mission via eve-survival site reports. What to shot in what order to finish the mission and not get screwed by triggers.
Start with sentry drone sniping Dominix
[Dominix, PvE cheap]

Drone Damage Amplifier II
Drone Damage Amplifier II
Drone Damage Amplifier II
Drone Damage Amplifier II
Photonic Upgraded Co-Processor
Damage Control II
Shadow Serpentis Large Armor Repairer

Large Micro Jump Drive
Omnidirectional Tracking Link II, Optimal Range Script
Omnidirectional Tracking Link II, Optimal Range Script
Omnidirectional Tracking Link II, Optimal Range Script
Sensor Booster II, Targeting Range Script

Tachyon Modulated Energy Beam I, Imperial Navy Radio L
Drone Link Augmentor II
Tachyon Modulated Energy Beam I, Imperial Navy Radio L
Tachyon Modulated Energy Beam I, Imperial Navy Radio L
Drone Link Augmentor II
Drone Link Augmentor II

Large Drone Scope Chip I
Large Drone Control Range Augmentor I
Large Sentry Damage Augmentor I

Warden II x5
Curator II x5
Bouncer II x5

Warp, deploy MTU, tank till you get into armor, than large micro jump 100 km from NPCCP.
Use your guns to kill fast small paper - thin targets that burns toward you with 0 transversal, so your drones keep working on large targets. After LMJD cools down jump back to gate and finish what few NPC is left. Range is your tank, NPC do not scram - they disrupt, so you always have your LMJD GTFO button with you.
After you learn missions you can decide what is more fun for you :
Full room clear
[Rattlesnake, L4 Whole room clean RHML]

Co-Processor II
C3-X ‘Hivaa Saitsuo’ Ballistic Control System
C3-X ‘Hivaa Saitsuo’ Ballistic Control System
Caldari Navy Ballistic Control System
Imperial Navy Drone Damage Amplifier
Imperial Navy Drone Damage Amplifier

Large Micro Jump Drive
Sentient Sensor Booster, Scan Resolution Script
Gist X-Type X-Large Shield Booster
Missile Guidance Computer II, Missile Precision Script
Missile Guidance Computer II, Missile Precision Script
500MN Quad LiF Restrained Microwarpdrive
Heavy Capacitor Booster II, Cap Booster 800

Rapid Heavy Missile Launcher II, Caldari Navy Scourge Heavy Missile
Rapid Heavy Missile Launcher II, Caldari Navy Scourge Heavy Missile
Rapid Heavy Missile Launcher II, Caldari Navy Scourge Heavy Missile
Rapid Heavy Missile Launcher II, Caldari Navy Scourge Heavy Missile
Rapid Heavy Missile Launcher II, Caldari Navy Scourge Heavy Missile
Drone Link Augmentor II

Large Drone Scope Chip I
Large Drone Control Range Augmentor II
Large Hydraulic Bay Thrusters II

Bouncer II x2
Wasp II x2

or brawler mission blitz:
[Machariel, L4 Mission blitz]

Gyrostabilizer II
Gyrostabilizer II
Gyrostabilizer II
Armor Explosive Hardener II
Corpus X-Type Large Armor Repairer
Energized Adaptive Nano Membrane II

Tracking Computer II, Optimal Range Script
Tracking Computer II, Optimal Range Script
Tracking Computer II, Optimal Range Script
500MN Quad LiF Restrained Microwarpdrive
Heavy Capacitor Booster II, Cap Booster 800
Large Micro Jump Drive

800mm Repeating Cannon II, Republic Fleet EMP L
800mm Repeating Cannon II, Republic Fleet EMP L
800mm Repeating Cannon II, Republic Fleet EMP L
800mm Repeating Cannon II, Republic Fleet EMP L
800mm Repeating Cannon II, Republic Fleet EMP L
800mm Repeating Cannon II, Republic Fleet EMP L
800mm Repeating Cannon II, Republic Fleet EMP L
Small Tractor Beam II

Large Hyperspatial Velocity Optimizer II
Large Hyperspatial Velocity Optimizer II
Large Hyperspatial Velocity Optimizer II

Hobgoblin II x5
Wasp II x4

More info can be found here:mission blitz guide