How to assign File/Folder path to a Variable in Terminal

Say I have the following path:

/home/$USER/Downloads/My Folder

I tried to assign the above given File-Path to a variable FILE_NAME as follows,

FILE_NAME=/home/$USER/Downloads/My Folder
$FILE_NAME

The output was:

bash: /home/$USER/Downloads/My :No such file or directory

How to resolve this issue?

You have to use quotes if the path contains space characters:

FILE_NAME="/home/$USER/Downloads/My Folder"

The issue is the embedded blank in the name. The simplest way to resolve this issue is to enclose the full path string with quotes (i.e. FILE_NAME="/home/${USER}/Downloads/My Folder"
The reason to use " in your case is because of your use of $USER which requires a substitution, with ' this would not occur.

A secondary question is how are you going to use the variable. In your example... I would assume that you dropped the cd from the command however ... to use the variable ... you should probably also use " around its use

so ... my guess at your use ... cd "$FILE_NAME"