although this solution is not the best and the safest solution exists, this worked for me and that make sense because all the problem we had with the errors is the permission problems. so you can use this command and ignore those permission problems :

sudo npm install expo-cli -g --unsafe-perm

This is a permissions issue. you can use sudo to fix this but as a general recommendation never run npm install with sudo because it is a bad idea in general ,you can read more about this here

So To minimize this issue you can configure npm to use a different directory for global packages installations as mentioned in npm docs here

1. On the command line, In your home directory, create a directory for global installations:

   mkdir ~/.npm-global
   

2. Configure npm to use the new directory path:

   npm config set prefix '~/.npm-global'
   

3. In your preferred text editor, open or create a ~/.profile file and add this line:

   export PATH=~/.npm-global/bin:$PATH
   

4. On the command line, update your system variables:

   source ~/.profile
   

5. Finnaly try to install expo-cli globally :

   npm install -g expo-cli 
   

Or if you want a quick fix you can try this

1. open a terminal and run :

   sudo chown -R $USER:$USER /usr/local/lib/node_modules
   

2. try to install expo-cli , run :

    npm i -g expo-cli  
   

Also consider to avoid globall installs and use npx when possible read more here

For Yarn:

If you want to save expo-cli package globally then try

yarn global add expo-cli

and if you want expo-cli to be available in the folder in which you are currently working then use

yarn add expo-cli

For npm:

npm install expo-cli --global

or

npm install expo-cli