Mass Effect 3 - The end of Trilogy 1 - The Shepard Cycle
Though there are different flavors of each of the three basic endings it is important going forward to focus on the three basics and how they will affect the future.
Destruction - Shepard may live or die depending on war assets. EDI dies, the Geth asplode and tremendous damage occurs to the entire tech base of Galactic civilization. The Citadel is likely also destroyed or in terrible shape. No more Reapers (though maybe some still exist in dark space for all we know).
The future: The Cycle is over. Reapers (at least in this galaxy) are dead and the rest are without the guidance of the citadel's systems...perhaps they will appear someday if they exist at all. The Galaxy will be rebuilding and whatever the main protagonist does it will be as an N7 operator.
Given that ending this game should have the most news about rebuilding and recolonization. It is also possible that the Krogan are dying if you chose to sabotage the genophage cure. The Geth are extinct and the Quarians are probably a non-factor having most likely concentrated on rebuilding their homeworld. With no Geth resistance they are free to rebuild...but without the geth help they COULD get in other endings they are probably still in poor shape and probably not a part of this trilogy in any manner.
The Geth...dead
The Krogan...dead or dying (perhaps still a chance to save them as we know for sure they are in Future Mass Effect).
Humans - Duh we're so vain that it acts as actual plot armor against extinction
Asari - yeah they should be around being one of the most advanced and wide spread civilizations
Batarians - Most likely extinct. Even if they are still around in some minor fashion they are a non-factor
Raloi - Perhaps the Reaper War passed them by since we never saw any of them ascended for Shepard and crew to kill.
Turians - In bad shape...most of their fleet destroyed and their planet cratered by the reapers...like Earth it would be in recovery but still have massive scars. Also a long time space-faring race they will be wide spread like the Asari.
Salarians - In the best shape probably. They are the second longest space-faring race of this current cycle and they would appear to have suffered little to no damage to their home world during the Reaper War. Possible repercussions with the Krogan. Would also put them at odds with Turians politically speaking.
Quarians: They could also be dead
Control - This is Shepard's favorite store on The Citadel! Shepard's ghost now controls the Reapers...making them a protective force and they probably helped rebuild everything. The Reapers are still giants astride the galaxy and should be seen around helping and protecting.
The galactic infrastructure should be beyond what exists - but perhaps people only criticize Shepard's parental control quietly. He might not be the kindest of despots. All of the council races should be closely tied and cooperating.
Humans - Earth rebuilt - probably now has its own reaper protection force
Turians - Palaven rebuilt - Probably with its own force of reapers to protect it
Asari - as above for Thessia
Salarians - Had few Reaper problems...probably now have Reaper guardians as well anyhow.
Krogan - could be dying still...or genophage cured
Raloi - who the fuck knows
Quarians - Perhaps they're dead - if alive they are rebuilding rannoch...if the geth are also alive the geth are helping them rebuild...and perhaps they chose to leave the galaxy behind.
Batarians - perhaps they have been able to rebuild with Shepard's help.
EDI likely survived as did the rest of your comrades - hey maybe you got Liara preggers!
Synthesis - The 'best' ending you can hope for. Requires the most war assets and requires you to have both deleted the geth heretics and blown up the reaper base beyond the omega 4 relay. The citadel is probably somehow now self aware...or at least self maintaining. The Keepers are likely released from their servitude somehow. Galactic infrastructure is not merely on par but likely above where it would be under the control option.
This best ending means everyone survived and is now enhanced...all life in the galaxy is now enhanced not just that which had direct contact with the Reapers. Life has now truly ascended. Beyond visual distinctiveness there should be some manner of other advancement seen.
Humans - Earth not only rebuilt but in a golden age
Asari - same as Earth
Turians - same as Earth
Salarians - Probably also in a Golden Age
Krogan - genophage cure being required for this ending...they are also in a golden age and Tuchanka is blooming. All past transgressions are put aside.
Raloi - Perhaps mistrustful (if they are in the game) since they have been forcibly altered by Shepard.
Geth & Quarians - Rannoch is rebuilt and the joint Quarian/Geth society is ascendent. Perhaps they are in a period of isolation or perhaps have fled the galaxy together.
Batarians - probably still rebuilding but better off than in any other scenario. Perhaps we get a trip to Khar'Shan?
In all of the scenarios it is likely that we have moved beyond the use of Mass Effect Relays to explore the Galaxy in a wider and more flexible manner.
My hopes - choices mean something. And endings mean something. Far enough into the future and the differences even out...but there would be massive changes at least in dialogue if not the actual appearance of the game!
Welp