Welcome to the PC Guide Forums. This is a great place to get help.
This is not a totally uncommon problem, so I'll respond here so others can benefit from a discussion of this. I mean, that's the purpose of these forums.
First, it would help if you would describe your system, whether you bought it, built it yourself, the components in your system. Did this suddenly start, or has the system always been this way? All of this information is important in addressing your question.
I've had similar problems at times like yours, and it had to do with running my system at very high performance settings. Specifically, my kt-600 system will not "restart" from within windows without resetting several of the bios settings upon boot. The system runs very stable, but I have it running at a very high cpu speed, memory settings, and so forth. If I relax those settings, i.e., set my cpu speed (fsb X multiplier) to run slower, then it restarts just fine. Plus, the system has been that way from day one. Also, when I had it running at its absolutely fastest speed, it would not even start from the front panel switch. I got the same identical symptoms you are describing, and had to turn off the psu switch, wait 10 seconds, and then turn it back on. My kt-333 system used to have similar issues. In contrast, my nf2-400 ultra system always restarts, though that might be due to it using a different psu. So your hardware and bios settings can affect this.
Others can better explain, but I think your system's failure to start-up the monitor has to do with the video card initializing, or something like that. I avoided the problem by using the master power switch on the psu , and doing so is in no way unsafe. However, it does cause wear and tear to that switch, and one way to avoid that is by using the on-off switch on a power strip or surge protector or uninterruptible power supply that is connected to your system.
Finally, if your system had been stable and then this suddenly started, well, that's a whole different story and would be cause for concern. That's why it is necessary to get more information.