For countably infinite graphs you can take the Rado Graph, which contains all countable graphs (even as induced subgraphs). For higher cardinals, see this paper of Shelah.

For countably infinite graphs you can take the Rado Graph, which contains all countable graphs (even as induced subgraphs). For higher cardinals, see this paper of Shelah. As I mention in a comment, the version with minors is not that interesting, since you can just take the complete graph on $\kappa$ vertices.