Is there a standard way to install the same programs across different linux systems in an automated way (a single script that would work on each system)? Most of my systems are Ubuntu and an Ubuntu-only solution that would install the same program across different versions of Ubuntu, where package names can vary for different versions, would work for 90% of my needs if you have one. I can think of a number of ways I can solve the problem but I thought I'd ask here first since I'm sure the problem has been dealt with before and your help could steer me away from potential pitfalls.
There is no a ’standard’ way to do it. You should consider some configuration management tool. The task that you describe is easy to do with an ansible playbook for example.