help update Redmine to the latest stable version

Support for Ubuntu 14.04 has already ended (except for paid Extended Security Mantenance for selected packages on server systems).

Furthermore Ubuntu is no rolling release - see FAQ #3037: “no rolling release”.

If you want an updated version of redmine, then the only supported way is upgrading your Ubuntu release.

OK, to which version of ubuntu should I upgrade?

The version of Ubuntu to which you should upgrade depends among others on the purpose of your system.

If you want to keep the system stable for a longer period, then you should stay on LTS (Long Term Support) versions.
If you want the newest available version of software, then you have to use also the non-LTS versions, but that requires a release upgrade every 6 months.

My suggestion is an upgrade to Ubuntu 18.04 LTS, with redmine 3.4.4
If that version of redmine is not high enough for your requirements, then you have to upgrade to Ubuntu 19.10 (with redmine 4.0.4). From Ubuntu 19.10 you will be able to upgade to 20.04 LTS next year.

This version suits me (18.04 LTS, with redmine 3.4.4)
Tell me where to download the distribution
And most importantly, how to transfer data from the old redmine to the new

You have the choice:

Either do a new installation, with a download of the installer from https://ubuntu.com/download
or do an in-place upgrade (with the command "sudo do-release-upgrade").

command "sudo do-release-upgrade" unfortunately executed with errors and does not end
I installed a new version of Ubuntu 18.04.3 LTS
How can I install the latest stable version of redmine and migrate data from the old redmine

For installing redmine 3.4.4-1ubuntu0.1 on Ubuntu 18.04 LTS just install the redmine package from the Ubuntu repositories, e.g. with the command "sudo apt install redmine" (and/or "sudo apt install redmine-mysql" and/or "sudo apt install redmine-pgsql" and/or "sudo apt install redmine-sqlite" depending on the database connectors that you intend to use). You need to have the "universe" repository activated.

For data migration I cannot provide help. I assume that you have to setup redmine and mysql in a similar way to the old installation, and probably have to copy the data files from the old system to the new one.

Is mysql installed or should it be installed in a separate package?

root@redmine2020:/home/admivc# dpkg -l | grep redmine
ii redmine 3.4.4-1ubuntu0.1 all flexible project management web application
ii redmine-mysql 3.4.4-1ubuntu0.1 all metapackage providing MySQL dependencies for Redmine

And also a question about installing Apache

You have to check yourself, whether mysql is installed on your system.

What is the output of the command

dpkg -l | grep mysql

(Remark: Installing redmine-mysql will also care for installing some mysql packages as dependency, but I do not know whether that is sufficient for your needs.)

# dpkg -l | grep mysql
ii dbconfig-mysql 2.0.9 all dbconfig-common MySQL/MariaDB support
ii default-mysql-client 1.0.4 all MySQL database client binaries (metapackage)
ii libmysqlclient20:amd64 5.7.28-0ubuntu0.18.04.4 amd64 MySQL database client library
ii mysql-client-5.7 5.7.28-0ubuntu0.18.04.4 amd64 MySQL database client binaries
ii mysql-client-core-5.7 5.7.28-0ubuntu0.18.04.4 amd64 MySQL database core client binaries
ii mysql-common 5.8+1.0.4 all MySQL database common files, e.g. /etc/mysql/my.cnf
ii redmine-mysql 3.4.4-1ubuntu0.1 all metapackage providing MySQL dependencies for Redmine
ii ruby-mysql2 0.4.10-0ubuntu5 amd64 simple, fast MySQL library for Ruby

it remains to find out about Apache

You will probably need to install the package apache2 and configure it similar to what the old system was.