Basically, the red Power button, the Volume up and down, and Mute buttons are "TV controls". They will only control your TV, no effect on the Roku device.
If you do not have CEC enabled on both the Roku device and also on your TV (located in your TV HDMI settings), then the 1-touch play feature will not work.
If not using "1-touch play" (any button press other than using the TV controls mentioned above) to wake up Roku device and turn on TV simultaneously, then pressing the Home button on your remote should wake your device. (pressing the Home button a few times won't hurt anything and may be required).
If not using 1-touch play, you will need to press Home to wake up Roku device, and then press the red Power button on your remote to turn on your TV. (or you can press Power first, then Home button. Order does not matter).
If the above does not help, try disabling "Power Savings" if you have a newer Roku model. This setting is located under Settings/System/Power/Auto Power Savings/20-min inactivity [uncheck to Disable].
Feel free to post back if still having issues and describe how you are/were trying to turn on the device and what was happening. Include Roku model number.
Roku Community Streaming Expert Just another Roku user... I am not a Roku employee. Insignia RokuTV, Ultra 4660, Premiere+ 3921, Express 4k+ 3941, Streambar 9102