The film would focus around the idea that the old, original Ghostbusters are indeed passing the torch. Seeing as how the previous two films have always used a subplot that they weren't exactly the most well-received and well-funded organization, that would apply here as well. At its core, Ray Stantz was always the driving force behind the group and this would continue. He's been trying to convince the others that the downtime is over with and they should look into restarting the group. Winston Zeddemore, the hard worker of yesteryear, is retired and likes it that way. Egon Spengler agrees that their days as cowboys are over with and is simply more interested in following up on the academic side of things. He feels that they're better served in that regard, particularly after the death of Peter Venkman. It just wouldn't be the same without the man who kept them all together.
Seeking to reignite their passion, Ray Stantz approaches Oscar Barrett, son of Dana Barrett from Ghostbusters II, with an offer to join them and become a Ghostbuster himself. The reason why is because over the course of the past 20+ years, Oscar (nicknamed Oz or Ozzy) was eventually told that Peter Venkman is his actual biological father. Oscar is very similar to Peter in terms of characterization (like father, like son). He is crude, cocky, antagonistic, and has just the right amount of charm to make people withstand him. Just like Peter, he comes off worse than he really is and is hiding a bigger heart than you'd assume. Oscar has no interest in joining Ray and the Ghostbusters, though. This is primarily because his relationship with Peter was very strained and he resents him for not fixing it before his death.
While Oscar might not be interested in joining up with the team, there are two people that do in fact want to be involved. One of them is a college student who has been studying under Egon for some time. He's a stereotypical nerd in pretty much every way and is very much Egon's successor already, being rather humorless and serious. Though this guy is more so supportive in following in Egon's footsteps instead of doing the physical side of things, he's nonetheless eager to be a part of the new squad.
Another person that wants to be involved, almost desperately, is quite the fanboy of the Ghostbusters. He's someone that has followed them for quite a while and thinks it would be cool to be a part of it. The problem is that he's a bumbling and inept idiot that seems to offer nothing (although in terms of storytelling, he's very much an anchor of comedic relief).
At the moment, I haven't quite figured out what the primary antagonist could be, other than the idea that it would revolve around two other characters. The first of which is a man who has some sort of political power and completely against the Ghostbusters. Ideally, this would be a returning Walter Peck (portrayed by William Atherton in Ghostbusters I). The other one of these two characters is his daughter. The backstory of this duo is that the daughter (hereby referred to as simply "Peck's Daughter") and Peck don't get along. Peck's dislike of the Ghostbusters has resulted in a stereotypical lashing out from Peck's Daughter in terms of personality. She's grown up as someone who is interested in witchcraft. There can even be a joke thrown in there that she was raised being unable to eat marshmallows because of the Stay Puft Marshmallow Man in the first film.
As if it isn't obvious, the four new members of the team are Oscar, Peck's Daughter, the Inept Fool and Egon's Apprentice. Ray acts as the overseer of the transition between the former squad and the trainees. Egon naturally is the head of the scientific side of things. Winston plays the role of their physical trainer, teaching them how to use the equipment for tactical purposes. Their first test trial is to capture Slimer, who has been utilized over the years as their sort of pet.
Each character of this new squad is there to counteract the others. It's self-explanatory that Egon's Apprentice clashes with the Inept Fool because of their intelligence. Peck's Daughter turns out to be the absolute best in terms of physical trapping of the ghosts, which puts her at odds with the Inept Fool because of their difference in quality. It also puts her at odds with Egon's Apprentice because they're on opposite sides of their quality, with him being the scientific end and her being the physicality. Oscar clashes with the Inept Fool because of their disagreement on whether they'd like to be a part of the team. While Oscar is reluctant, the Inept Fool considers it blasphemy that Oscar isn't following in his father's footsteps. Oscar's antagonism to Egon's Apprentice is about dedication as well but more so about the idea that Egon's relationship with his apprentice is what Oscar would have liked for himself and Peter.
The biggest relationship between the squad, however, is the connection of Peck's Daughter with Oscar. They're very competitive towards one another but they're also very supportive because they can bond with one another over their father issues. That bond is also a romantic relationship. In fact, the main reason that Oscar is convinced to become a member of the squad is because of his lust for Peck's Daughter.
By the end of the film, some of the following things happen. The Inept Fool turns out to be rather useful, possibly as a means of funding the group monetarily (on top of being the heart of the new group). Louis Tully is able to defend the Ghostbusters as their lawyer (with the help of his wife and secretary, Janine) reducing their roles to a cameo. Peck's Daughter, with the help of Oscar, is able to tell Peck to go to hell, resolving many issues they both have for their fathers. This applies even more so to Oscar when it comes to a cameo of Peter Venkman's ghost who not only helps the Ghostbusters out, but also has a conversation with Oscar. They hash things out. Oscar admits that the main reason he was acting the way he was acting is because of him missing Peter and wishing he hadn't left him, as he loves him. Peter admits that he pushes people away and ultimately makes it hard for them in some ways. This happened with Dana, but they've always loved and forgiven each other no matter what. He lets Oscar know that Peck's Daughter and his relationship reminds him of his relationship with Dana. He also lets Egon, Winston and Ray know that they have this covered and his full support, along with the rest of the team.
The old school team is forced to suit up one last time to help the new school team with the paranormal threat, whatever it is. Peck's Daughter's knowledge of witchcraft proves to be particularly helpful and they're even aided by Peter and Slimer from the other side of things. When all things are said and done, they've defeated the evil, become a solid team, and gotten over their issues together. You can either leave this open-ended, that they'll continue being the new team or you can go ahead and do Ghostbusters IV afterward.
That being said, who you gonna call for help on this new film? My line's open if you like the outline above.