The first few missions are pretty linear until you leave the citadel for the first time, but you will realise pretty quickly how they have to be.
PROLOGUE EARTH - No companion choice.
PRIORITY MARS - Companions not optional.
PRIORITY THE CITADEL - Companions not required.
Once you're back on the Normandy speak to everyone you can speak to. Also, if you bought all of the ship models in ME2, you can find them dotted around the ship. They are mostly in the cargo bay and the storage under engineering, but one is in Thane's old room.
RESCUE TURIAN PRIMARCH (Scan the surrounding systems in the cluster.) - James and Liara.
DOCK AT THE CITADEL
I'd like to give you a recommended order of which elevator platform to choose at any given time, to ensure that you do not miss anything because this first citadel trip is a doozy. This list is by no means detailed in an attempt to be as spoiler free as possible but there could be people to speak to/things to do on each floor. Once you've spoken to someone check the map to see if there's anything else you need to do.
Docking bay.
Presidium Commons. (If you want your fish to live, buy the aquarium VI)
Purgatory.
Embassies.
Huerta Memorial Hospital.
Bay D2.
Embassies. You should be in a position to complete CITADEL: HANAR DIPLOMAT. If not, check the map and catch up with the terminals.
Presidium Commons. A lot of people marked on the map, but there will be a character from ME2 behind the bar at Apollo's cafe that is not marked and very much worth speaking to.
Bay D2.
This should finish your first big trip to the Citadel. Go back to the Normandy.
SEARCH KITES NEST CLUSTER
MEET WITH DIPLOMATS - Garrus and Liara.
RECOVER PROTHEAN ARTIFACT - Liara (mandatory) and anyone (my personal favourite is Garrus)
RESCUE THE STUDENTS - EDI and Garrus/Liara.
CERBERUS LAB - Anyone.
SEARCH ISMAR FRONTIER CLUSTER
SEARCH KROGAN DMZ CLUSTER
INVESTIGATE CERBERUS PRESENCE - Anyone
DOCK AT THE CITADEL
DOCK BACK AT THE CITADEL
EXTRACT TURIAN SURVIVORS - Garrus and Anyone (it's a turian heavy mission.)
TUCHANKA BOMB - Garrus and Anyone.
INVESTIGATE MISSING SCOUTS - Javik and Garrus.
CERBERUS ABDUCTIONS - Anyone.
DOCK AT THE CITADEL
CITADEL: BARLA VON Should be completed by this visit, otherwise it locks out.
CITADEL: IMPROVED POWER GRID Should be completed by this visit, otherwise it locks out.
CITADEL: BIOTIC AMP INTERFACES Should be completed by this visit, otherwise it locks out.
Even though it looks like it should be, you will not be able to complete CITADEL: HEATING UNIT STABILIZERS and CERBERUS AUTOMATED TURRET SCHEMATIC at this point, so hang on to them.
PRIORITY: TUCHANKA (THE FINAL MISSION ON TUHANKA) - Take your favourites, everyone has unique dialogue.
Don't forget to speak to everyone.
CERBERUS FIGHTER BASE
SCAN ALL SYSTEMS THAT ARE NOT ALREADY AT 100% (Some will have nothing but it's worth just going through them and making sure you've collected everything you can.)
DOCK AT THE CITADEL
There is again, a lot to do. Hand in missions, talk to people, resolve a lot.
The two missions you couldn't turn in later should now be available to turn in.
A familiar face that appears across all three games should be in the refugee camp, if you've done everything in the previous games.
FIND EX-CERBERUS SCIENTISTS - Virmire Survivor/Garrus and EDI
SPEAK WITH QUARIANS - N/A
You might have some people to speak to on your ship.
INVESTIGATE ASARI COLONY - Liara and anyone.
DOCK AT THE CITADEL
Do not visit Anderson's apartment. We save that until the end.
This is the "solidify your romance" section. The dialogue is really better if you see your love interest first, otherwise you're going to have to let down a fair few people.
You will probably have more missions to hand in or pick up as well. Check the terminals in the Spectre room for anything you might have missed.
Use a sky car to visit Dr Bryson's Lab, and to start my favourite DLC of the entire game.
DISABLE THE DREADNOUGHT - Tali (mandatory) and your Love Interest. Or, if Tali is your love interest, the person you romanced in ME1. If you didn't romance anyone in ME1, bring EDI or James.
SCAN FOR DR GARNEAU - Your Love Interest and Anyone else (I like taking EDI because she's so present in the lab)
A friendly reminder to talk to everyone on your ship after each mission.
SCAN ANY SYSTEMS THAT YOU HAVEN'T ALREADY SCANNED AND ARE NOT AT 100%
FUEL REACTORS - Anyone
DOCK AT THE CITADEL
There are likely a couple of missions to hand in or take, not a lot though. Check the terminals in the Spectre room for anything you might have missed.
Go back to Bryson's lab once you've been to the main part of the citadel.
If you have been able to afford it, you should now have all model ships and all available fish.
SAVE THE ADMIRAL - Tali and EDI
SHUT DOWN GETH SERVER - Tali and Anyone
SCAN THE DIG SITE - Your Love Interest and Anyone
Another reminder to talk to everyone once you get back to the Normandy. There are so many fun interactions, not just between Shepard and characters, but in the conversations they have with each other.
DOCK AT THE CITADEL
You know the drill, missions, missions missions. Check the terminals in the Spectre room for anything you might have missed.
Go back to Dr Bryson's lab.
DESTROY REAPER BASE - Tali (mandatory) and Anyone
Talk to everyone back at the Normandy.
RENDEZVOUS WITH ARIA'S FLEET - Solo time, this is a big chunky bit of DLC.
DOCK AT THE CITADEL
LOCATE SECRET ARTIFACT - Liara (mandatory) and Javik.
Scan any systems you may not have already scanned, and talk to everyone on the Normandy.
SEARCH FOR LEVIATHAN - Your Love Interest and anyone.
COMMUNICATION HUB - Anyone.
INVESTIGATE SANCTUARY - Virmire Survivor (if they are an option) and Anyone.
DOCK AT THE CITADEL
Hand in missions, you shouldn't have any more to take.
Go back to the Normandy and immediately turn around and...
DOCK AT ANDERSON'S APARTMENT
Welcome to fan service, the DLC. It's brilliant and wonderful and this is why we leave it until now.
Get changed into something fancy before meeting with Joker.
Black tie required? Take either your Love Interest or Wrex.
Archives? Take anyone.
Citadel docks? Anyone is excellent. I love Garrus with anyone (there is a ME2 callback) and EDI has a really funny bit early on, but they're all so brilliant.
Citadel Extras. When meeting up with people there are three that are not one the map. Two involve waiting at the bar upstairs in the casino, and one is on the lower floor ("investigate suspicious device")
If you are romancing Garrus, opt out of the third and final interrupt in his section.
If you have anything left in the journal complete it now before PRIORITY CERBERUS HEADQUARTERS. This is the point of no return.
ASSAULT THE ILLUSIVE MAN'S BASE - EDI (mandatory) and your Love Interest.
ORDER FLEETS TO EARTH
Part one - James/Virmire survivor and Love Interest
Part two - Your Love Interest and anyone you want.