Usually there is no need to download packages manually that are listed in go.mod file. go command can automatically download all direct and transitive dependencies of the project in the module cache and then use those downloaded packages while building the project.
npm works differently since it downloads all direct and transitive dependencies of a project into a node_modules directory local to the project.
If you want npm like package management in Go, the closest you can get is with the Vendoring feature of Go.
go mod vendor
command creates a directory named vendor in the root of main module containing all the packages that are necessary to support the builds and tests of the packages in the main module.
More detailed information about Vendoring is available in official Go docs.