We will likely get the opportunity to see the court of Gondor in Minas Tirith this season - just a glimpse, a single episode frame, two 2-5 minute scenes, and that's it. Most likely, we will want to use that opportunity to establish a stark contrast with whatever is happening in Harad at this time.
So, what does Gondor's court look like?
Turgon is the aged steward. He is about 96 years old, and nothing eventful enough to wind up in the Appendices of LotR happened during his reign - so basically, it's been a long, peaceful stretch for him, likely focused on internal issues (though not much strife). His reaction to Gandalf's news (Sauron has returned! Harad is being brought to his side!) is likely to lean in the direction of...I can't really be bothered with that. He dies in TA 2953 (just after this). He is a good reason for Gandalf to say something along the lines of 'the war will not come in your lifetime, but in your son's or even grandson's.'
Ecthelion II, Turgon's son, will succeed him in a few years. He clearly *does* take Gandalf's warning to heart, and works on bolstering allegiances and foreign policy - he will later welcome Thorongil. He strengthened Gondor's defenses and prepared for war. Ecthelion is in his 60's during Gandalf's visit, and likely already responsible for much of the day to day running of Gondor as his aged father's heir. Gandalf can probably appeal to him with his responsibility as a steward.
Denethor (born TA 2930) is a young man, aged 20. We should see a lot of Faramir in him, I think, as Faramir is the one who truly took after him (despite his preference for Boromir). He should view Gandalf's warning with some skepticism - How do you know this? What were you doing in Harad, anyway? Doesn't anyone else find this a bit suspicious/convenient/alarmist? etc. He's not a villain, but he definitely does not wholeheartedly embrace Gandalf's message as a call to action. Gondor has been peaceful his whole life, but he knows history, and he's not naive.
Thengel, crown prince of Rohan, is a permanent resident of Gondor, and could easily appear at court. It will obviously be up to him to pass this news along to his father Fengel in Rohan. Although, spoiler alert, he and dad don't get along (at all), which is why he lives in Gondor with his wife and not in Rohan. He's in his late 40s at this point, and in only a couple years will be called back to Rohan to rule. So it will be up to him to respond to Gandalf's warning here. Does he? Theoden is a 2-3 year old child at this time, and will therefore likely not appear.
I imagine that the friendship of Thengel and Ecthelion should be viewed as a precursor to the friendship of Eomer and Aragorn. They both assume leadership at the deaths of their fathers at about the same time, and Rohan and Gondor seem to be getting along quite well with one another throughout their reigns. Ecthelion accepts Thorongil into his service in part because he came from Thengel's service.
Again, we're just going to get a glimpse of this court - we're not delving into all of this. Thengel is going to be the random blond dude who says something about 'my people' or 'Rohan' that makes it clear he's a foreigner. That's all. It's basically a teaser to set up 'The Adventures of Thorongil' later. But I hope that this snapshot will be a really good foil for what is going on in Harad, so wanted to put this down for reference as we come up with ideas for the situation in Harad at this time.