Sent from my HTC One_M8
Before you mess around i suggest to read up on terms and issues you might stumble on ! Learn adb/fastboot commands to the bone and then start all the rooting stuff.
Phone on :
Hold power + volume up, when the screen turns black (the reboot) you switch from volume up to volume down and keep the power button pressed all the time in this process. Release the buttons when you entered the bootloader.
Phone on and fastboot enabled :
Open a fastboot terminal on the pc and type :
Adb reboot bootloader
That should get you started.
For the step I highlighted in red, you mean power + vol up (forced reboot).
Power + vol down while phone is on takes a screen shot.
---------- Post added at 02:29 PM ---------- Previous post was at 02:23 PM ----------
It can be done with the phone on (reboot the phone) or off (truly off, not with "Fastboot" selected in Settings, where the phone is not truly shut off).
If you do it from a reboot (either forced, or just selected reboot after holding down the power button) you don't even have to deselect "Fastboot" in Settings (only required if trying to do it from a powered-off condition).
---------- Post added at 02:31 PM ---------- Previous post was at 02:29 PM ----------
If you searched specifically for the M8 methods to get into bootloader, they all should work. Either its just pilot error in doing the steps properly, or you are looking at instructions for other devices (which is suspected, since you reference "download mode" which is Samsung-specific).