You need to install mysql first, the log contains
mysql client is missing. You may need to 'apt-get install libmysqlclient-dev' or 'yum install mysql-devel', and try again.
The documentation page of the mysql2 gem contains:
This gem links against MySQL's libmysqlclient library or Connector/C library, and compatible alternatives such as MariaDB. You may need to install a package such as libmysqlclient-dev, mysql-devel, or other appropriate package for your system. See below for system-specific instructions.
By default, the mysql2 gem will try to find a copy of MySQL in this order:
* Option --with-mysql-dir, if provided (see below).
* Option --with-mysql-config, if provided (see below).
* Several typical paths for mysql_config (default for the majority of users).
* The directory /usr/local.
Try to install mysql or the mentioned libs and try again to install the gem.
The messages do not seem to reflect reality, try this instead of the mentioned package:
sudo apt-get install default-libmysqlclient-dev
Then, your next error clearly shows what is wrong
/home/pi/.rvm/gems/ruby-2.6.3/gems/execjs-2.7.0/lib/execjs/runtimes.rb:58:in `autodetect': Could not find a JavaScript runtime. See github.com/rails/execjs for a list of available runtimes. (ExecJS::RuntimeUnavailable)
The GH page says:
ExecJS lets you run JavaScript code from Ruby. It automatically picks the best runtime available to evaluate your JavaScript program, then returns the result to you as a Ruby object.
Install one of the javascript runtimes, listed here in the readme, e.g. nodejs (https://nodejs.org/en/download/), therubyracer, etc.