If we're going to disregard contracts and assume the veterans can be moved in deals that make sense, then the only player I'm comfortable projecting as a future core piece is Suzuki. Maybe Caufield, but I'm not so sure about that.
The way I see it, the core will be somewhat fluid in the coming years. Some veterans will be kept, because we have no one ready to fill those roles and/or they can't be moved due to contact considerations. Here is the core:
#1G: It's going to be Price until the end of his contact. It's impossible to trade him without seriously hampering yourself imo. Then, perhaps Primeau, but it's far from certain. Anyways, it's not really relevant to the rebuild, because teams don't hit a rebuild button in order to draft a goalie with a top-10 pick. They usually come from other sources.
#1D: Jeff Petry. Inadequate, but there's absolutely no one in the organisation who projects to fill that role. For this reason, and because of a lack of NHL ready RHDs, he might be here to stay. This role is the most glaring long-term need.
#2D: Joel Edmundson. Inadequate, but there might actually be players in the system to fill that role in the future, with Guhle being the frontrunner.
#1C: Nick Suzuki
#2C: Depending on how he turns out, and on Suzuki's progression, it might still be Dvorak. We'll see.
#1W: Gallagher. Even assuming that what we're seing today is just a bump on the road, it's not ideal if Gallagher is your best winger. Since we might have to keep Price's and Petry's contracts on the books, he's the most logical piece to trade, although it will have to be carefully considered due to chemistry and leadership reasons. The team has taken quite a lot of hits in this department in the recent months. As for the future, we don't really have a star winger in the system, but it's a relatively least important part of the core.
#2W: Toffoli then Caufield.
Whatever we decide as regards to Gallagher, I think Anderson absolutely needs to be traded this season or the next. He's an exclusively rush player with poor awareness and puck protection ability on the cycle. He's going to be absolutely useless once his skating ability diminishes. On a team full of risky contracts, he might just be the nail in the coffin, because he's completely redundant and would provide solid value if traded now. Even with Gallagher gone a RW depth of Toffoli/Caufield/Armia should be perfectly adequate. LW is the easiest position to fill, especially if we retain Lehkonen and perhaps even re-sign Drouin for less money to serve as a playmaking winger on Dvorak's line.
If I decided to trade Gallagher, my next season's team would look something like this:
xxx - Suzuki - Caufield
Drouin - Dvorak - Toffoli
Lehkonen - Evans - Byron
xxx - Poehling - Armia
Edmundson - Petry
Clague - Savard
Romanov - xxx
Price
Allen