If my computer needs to contact another host on a different network, and my computer did not have the destination host's mac address cached, would my computer send a broadcast ARP request, or would my computer not even bother and realize that the desination host is on a different network (based on my IP and subnet mask) and send the actual frame to my computer's gateway address so it can be routed to the appropriate network? I'm assuming it just sends it to the gateway address , but I wanted to be sure.
You are correct in that it will go to the default gateway. Your machine checks its subnet to see what network its attached to and if the destination is on a different network then it sends everything to the default gateway.
if you ping computer A from computer B, first the computer B does an anding function with its subnetmask inorder to find wether the destination is a local network or remote... if it is remote, then it forwards it to the gateway. In case if it is local then it looks in its arp cache for any previous entries ( so that it can contact without broadcasting ) If no previous entries, then it does a arp broadcast. which reaches every computer in the network and the particular computer with the requested ip will reply with it's mac address..
if you use a ethereal software , you can easily sniff the packets live