I recently installed the latest version of emacs and Rust-analyzer, and downloaded the lsp-mode package as well, but when I activate the rust-analyzer with the "lsp" command it simply returns an error saying that the rust-analyzer keep loading
rust-analyzer can take a while to index and load a project. You should still be fine, even if the text in the minibuffer doesn't immediately change. Try using e.g. the auto completion and see if it works.
If you want to test a prebuilt config, I recently written a guide to setup Emacs for Rust and it has a repo with an example config that you can run as-is. This was tested with Emacs 27.1, MacOS, Win 10, and Ubuntu.