Zing, connect as closely and as direct to the battery as you can get, and have that line fused somewhere close to the battery also.
As for figuring out how much fuse you'll need, just divide the watts by 12 to calculate the amps.
Power (watts) = amps x volts,
or another way of looking at it,
amps = watts / volts.
That means if you could possibly draw 1000W from that inverter (and forgetting about any inverter inefficiency) then the battery draw would be about 83 amps! Far more than ever could be drawn from the 12v taps on the dash. Don't even consider it, unless you are only using something less than 300 watts (25 amps at that!).
Even that 720W draw, you had mentioned being just 30A, can't be right. My guess is that while the drill motor you plugged in was a 720W motor, it was not using that much power at the time. It probably was unloaded. 720W figures out to 60A at 12V.
For handling 80 amps you should use pretty hefty wire, like 6 gauge. Here is a chart that will assist you in determining how heavy of wire to use:
Amps and Wire gauge
Good luck, and don't burn up your rig.