What hotkeys should a programmer know?

Ever tool has different HotKeys.

Start with your specific tool's default bindings, then just change them as required.

Your unique workflow will determine which HotKeys you use most.

I remember... way back... around 1980ish... when I was getting started with vi, each time I ended up doing a repetitive task, I'd just create some random HotKey.

My guess is most people have a unique way of coding + typing, so likely ever person will be slightly different.

As @David Favor said, it depends on the apps you are using.

However, if you are programming under Windows, start by remembering the Windows hotkeys.

If your app uses the traditional Windows menu system (the old File, Edit, etc system, not the new systems first introduced with Office 2000 and since bastardised by many apps), you should find the shortcuts showing in the menu. For example, clicking on File should show Save as Ctrl-S, Print as Ctrl-P, etc. In addition, you can always access the menu with the Alt key. For example, the key sequence Alt-F-S will save your file, Alt-F-P will print it. In many apps the relevant keys are underlined when you open the menu. Note that these keys are app dependent. For example, Alt-F-A usually means "Save As" but in Irfanview it brings up the scan window.