I take the initial movements of the three houses of the Edain to look something like this:
View attachment 2712
According to the published
Silmarillion, the meetings of the Edain and Green-elves varies slightly by house.
The Green-elves complain to Finrod that the Men of the House of Beor are "'hewers of trees and hunters of beasts; therefore we are their unfriends, and if they will not depart we shall afflict them in all ways that we can.'" There is no evidence one way or another as to whether the House of Beor met the Green-elves, but the Green-elves clearly observed their hewing and hunting. I do not believe the Green-elves attacked these Men, because they say that they will "afflict" them, not "continue afflicting" them.
The Haladin, "meeting the unfriendship of the Green-elves they turned north and dwelt in Thargelion." This suggests there was some interaction between these Men and the Green-elves. Whether it was a physical attack or not depends on how we want to interpret "unfriendship," but I do not think we should have the Green-elves and Men actually kill one another.
Marach and his people (the House of Hador) "were a tall and warlike folk, marching in ordered companies, and the Elves of Ossiriand hid themselves and did not waylay them," so I do not believe they had any interactions with the Green-elves at all.
As for meeting the Dwarves, the published
Silmarillion does not indicate whether this happened or not. Based on my map above, they do go near Nogrod and Belegost, and it is explicitly mentioned that people of Marach "came down the Dwarf-road." Whether and how we want Men and the Dwarves of Nogrod and Belegost to meet and whether that meeting comes before or after Men cross into Beleriand seems to be up to us.
The meeting with Dwarves in northern Rhun mentioned here is, I believe, derived from the following quote in
The War of the Jewels: "In ancient days the Naugrim dwelt in many mountains of Middle-earth, and there they met mortal Men (they say) long ere the Eldar knew them; whence it comes that of the tongues of the Easterlings many show kinship with Dwarf-speech rather than with the speeches of the Elves." I may be missing it, but I saw no mention of northern Rhun or the Edain specifically in the chapter cited in Tolkien Gateway. It is certainly possible that the Edain met Dwarves, but I think it is also possible that the Edain did not meet Dwarves outside of Beleriand and only the Easterlings, who moved west later, met the Dwarves and had their language influenced.
Based on the published
Silmarillion, the only group the Edain definitely met before they came into Beleriand was "the Dark Elves east of the mountains."