Hi Robert,
There are a couple of things about your setup.

First, a 7 cell battery pack is 8.4Volts ( each cell is 1.2Volts ).

Second, the 8.4V battery pack that you have might be a bit more than the Speed 300 motor can handle long term. If you're using the 6V version of that motor, it's only rated up to a 7.2V battery pack. Beyond that your going to burn the motor brushes up pretty quickly and will have to replace the entire motor. Consider returning this and getting a good 7.2V NiMH.

Third, Speed 300 motors can draw upwards of 8+ amps at full power. The actual amp draw will depend on which 300 you have, but I think there are only two different kinds (a 6V and a 7.2V), sold under a half dozen labels (GWS, Graupner, Simprop, etc). Regardless, the 5 amp ESC you have will go poof soon.

Fourth, if you have the battery connected to the ESC, the ESC connected to the motor, and the ESC connected to your receiver, the receiver should be getting power without arming the ESC. If it isn't then your ESC is already toast and should be returned / repaired. It sounds as if either you connected the battery to the motor out cable; or you had it connected the correct way and ran the motor up to full power which then overloaded it (see #3 above).

I haven't had the best luck with Great Planes ESC's. They use garbage wires and connectors have little if any documentation, and their support is lacking. You might consider purchasing either a Pixie-7P ($35) or a Pixie-20P ($40) from Castle Creations (http://www.castlerc.com). They are very light, have rock solid performance, and great support.

I'm somewhat surprised that the motor even started if you didn't have a battery connected directly to the ESC. This is another sign that there is something wrong with it.

Also, there are only two basic types of ESC's: Brushed and Brushless. The primary difference is what kind of motor you can connect to them. The Speed 280, 300, 400, xxx motors are all brushed. Brushless motors typically come from Himax, Astro Flight, Aveox, Jeti, and others and are quite expensive ($50++) Of course, this doesn't get into the various options that each ESC on the market has such as switching rate, BEC, brake, programmable LVC, etc.

Good luck to you,
chris.