Between each mission talk to your squadmates to see if they have anything new to say. This speeds up their loyalty missions unlocking and ensures enough Paragon/Renegade points to unlock the necessary Charm/Intimidate options when they reach them.
1) Normandy SR-1
2) Lazarus Project
3) Freedom’s Progress
4) Normandy SR-2 – Wander the ship to meet and greet your new crew and accept some side missions.
5) Citadel – Visit to meet Anderson, buy some upgrades and do some side missions. Recruit Kasumi if you have the DLC.
6) Omega – Meet with Aria, buy some more upgrades and do some side missions. Recruit Zaeed if you have the DLC.
7) Recruit the Professor (Mordin)
8) Kasumi’s loyalty mission (non-Soldier classes only) – Non-Soldiers rely heavily on SMGs. Doing this mission early gives you access to the Locust, the game’s best SMG.
9) Recruit Archangel (Garrus)
10) Recruit the Convict (Jack)
11) Mineral Scanning/Side Missions – Take a break to do some mineral scanning. Talk to Miranda to upgrade your scanning systems to make this easier. To research all upgrades throughout the game you need 33,000 Element Zero, 267,000 Iridium, 265,000 Platinum and 340,000 Palladium. Add an additional 100,000 to each of these before you complete the Suicide Mission to get a bonus to a war asset in ME3. Do a few side missions along the way like the Firewalker and Normandy Crash Site DLC.
12) Recruit the Warlord (Grunt) – Be sure to visit Grunt in the hold afterwards.
13) Horizon – Mandatory.
14) Recruit the Assassin (Thane), ask about ship upgrades.
15) Recruit Tali
16) Illium – Buy upgrades, do side missions and speak to Liara.
17) Lair of the Shadow Broker – DLC mission. The Shadow Broker ship will be unlocked upon completion and offers a number of benefits, including the ability to respec followers. ~ Credit goes to BarrinVorg in comments.
18) Recruit the Justicar (Samara)
19) Mineral Scanning/Side Missions – Take a break with side missions and mineral scanning to buy all the upgrades you’ve found in the game so far.
20) Grunt’s loyalty mission – After completing it you get a discount at Fortack’s shop.
21) Mordin’s loyalty mission
22) Collector Vessel – Mandatory. Automatically starts after you complete five missions after Horizon. Do not go get the IFF yet.
23) Kasumi’s loyalty mission (Soldier players)
24) Miranda’s loyalty mission
25) Jacob’s loyalty mission
26) Garrus’s loyalty mission
27) Thane’s loyalty mission – Can be failed. Make sure you get the info during interrogation and don’t lose track of your target.
28) Jack’s loyalty mission – Once it is complete Miranda and Jack will have a fight. You need to use Charm/Intimidate to stop the fight and keep both their loyalties.
29) Samara’s loyalty mission – Can be failed. Be sure to lure your target out, never mention family or Justicars and remember all the stuff you learned while investigating.
30) Mineral Scanning/Side Missions – Take a break with side missions and mineral scanning. At this point you’ll have unlocked every character except Legion, and you’ll have unlocked the Advanced Weapons Training at almost the earliest possible point.
31) Tali’s loyalty mission – Can be failed. Do not present the evidence. Use Charm/Intimidate for the best conclusion or Rally the Crowd if Veetor and Kal’Reegar are around.
32) Zaeed’s loyalty mission – Can be failed. Renegade players will have no problem. Paragon players need to pass a very high Paragon check.
33) Overlord – DLC mission.
34) IFF mission – If you want everyone to survive the Suicide Mission, treat this as the point of no return. Do all your mineral scanning and visit the Citadel, Omega, and Illium one last time to finish all side missions and buy more upgrades.
35) Normandy – Immediately recruit Legion in the ship’s AI core behind the med bay. Talk to him again to get his loyalty mission.
36) Legion’s loyalty mission – Bring Tali for some bonus dialogue. Afterwards Tali and Legion will get into a fight much like Miranda and Jack. It is imperative that you use Charm/Intimidate here to make peace between them.
37) Normandy – Mandatory. For this small mission you’ll be controlling Joker. Follow the lights and EDI’s instructions carefully.
38) The Suicide Mission – Go there immediately. Don’t stop anywhere else.
39) Arrival – DLC mission.
Original Link – Continuation of discussion