Keyboard bindings from bash to zsh?

Almost everything you can do in bash has a zsh equivalent, but you have to handle the translation on a case-by-case basis.

The line editor in zsh is zle. The command to bind keys is bindkey. The line edition commands that can be bound to keys are called widgets.

# You may want to call different history search commands, e.g.
# down-line-or-history or down-line-or-search (and up-*)
bindkey '^P' history-search-backward
bindkey '^N' history-search-forward
bindkey '\e[A' history-search-backward
bindkey '\e[B' history-search-forward
# The others should work already