On my 85 Rx7 12A GS over the last few days I have been noticing an issue when I turn the key. Seems like when I move the key foward the first time with the dash lights on there is nothing, no starter engaging. I pull the key back and move it foward again and it starts. Seems like its been on the second try every time the starter will engage and turn over. I had the starter replaced last fall and the rest of the ignition has been pretty much changed. Wires, plugs, cap rotor, coils, starter, fuel pump, filter, rebuilt carb. The battery was replaced a few yrs ago and it did die on me over the winter due to not running. I jumped the RX with my other car it has been starting and running until this issue recently. Could it be the battery going?

 

Its possible to have your battery going out, but I'd put my money on the key ignition switch or a fluke in your solinoid on the new starter.

Take a multimeter to your key switch... best way to find out ;-)