So, you've got a brand new PlayStation 4, but unfortunately you've got "No Signal" now, too. You can't really game or watch movies on your new toy if you can't even bring up the PlayStation menu on your TV screen.
However, there are a few ways to fix the no connection alert, and I'll cover the most common solutions below.
Some people, including IGN and the peeps over at Kotaku, have been able to fix the "No Signal" issue by fixing the HDMI cable port inside the PS4. If your HDMI cable is not flush with the case, you may have a bent piece of metal inside the port.
In the first image below, the cable is not all of the way in, compared to the second image, which is how it should look. This is the first thing you should check, since it's so obvious. If yours is like the second image, then you should skip down and try one of the other solutions.
If it's like the first image, check to see if there is a piece that is out of place inside your port. You can take a flathead screwdriver to push it down. You can also try another HDMI cable. For more information on this problem and the possible fix, check out Kotaku's video below.
Just as you would to troubleshoot a computer, you can launch your PS4 in safe mode in hopes of remedying the signal issues. With the PS4 off, hold down the power button for 7-8 seconds on boot to launch into safe mode (release the button after the second beep).
From here, you can choose to restart the system normally, change resolution, update system software, restore defaults, rebuild database, and initialize. You can learn more about what each of these does on the PlayStation Support site, but I recommend updating the software first.
If you don't have an Internet connection available on your PS4, you can also manually update the software via a USB flash drive.
But, it's kind of hard to update your software if you're still not getting anything on your screen. In that case...
If your PS4 is blinking blue, and never entering the powered state (white light), then it could be a power issue. Turn off the PS4 using the force-shutdown method of holding down for at least 7 seconds until the PS4 beeps twice. Disconnect the PS4's power cord from the wall outlet, then inspect the power cord for damage, including the pins where you connect it to the PS4 directly.
If there is any visible damage, that can be your culprit.
The PS4 requires an HDTV and only has a HDMI output, so the first way to troubleshoot is to make sure it's not your HDTV. If you have another HDTV around the house, give it a try and see if it connects.
Still a no-go? Try updating the firmware on your HDTV. If you have a smart TV with embedded internet, this shouldn't be too hard to do, and it has been known to fix the issue with some users.
Also, try switching out the HDMI cable, which could be damaged itself.