Would that mean each player/client has a connection to every other client? relaying out timestamped activity info?
No ! That would be a hell to synchronize everything .
You should either have a centralized server in a dedicated machine, or have one of the players to host a game (but that would lead to a lot of issues for people behind NAT) . Then all players connect to the server .
In both cases, each player send their actions to the server . The server then validates it and broadcast to other players .
In short, that's it .
Maybe you should try and create a simple chat application with the same architecture before moving on to your RTS. You'll learn a lot and anyway you'll probably want to reuse the code for an ingame chat .