My PC has two HDMI outputs. One is built in and the other is on my NVIDIA GeForce graphics card. I use the VGA output on the card for my monitor. I tried to also hook up my Toshiba flat screen (to watch Netflix, etc.) with an HDMI cable.

First I tried the built in port, but got no signal to my TV. Next, I tried the graphics card HDMI output with mixed success. The TV showed the desktop, but nothing else. When I opened a browser the TV did not mirror what was showing on my monitor, thus no Netflix on my TV... I am not sure if I need to change any of the NVIDIA settings or something on my TV...

I uses dvi to hDMI cord works fine. you cant use built in and the card at the same-time with a pc. another platform u can .back to the pc.
Your default is based on the video card . Unless you have 2 ports on the video card you cant show both the monitor and tv at once. you should have a pc aka vga port on the tv so vga port ,but my guess is the default would have to be changed to onboard video .That would disable your video card.. you dont want to do that.
top it all off the pc doesn't sync with the tv never will its the architecture of the pc. SO one will look off . Its the computers fault not you.