Thanks dannable for all your support and lucid explanations and illustrations. I have tried to use the connection based on the image you posted, but it did not work for me. I did not get any response when I typed in AT into my Serial Monitor.
The IDE I am using is ARDUINO 1.8.4 from https://www.arduino.cc/en/Main/Software, and I got word from the shop where I bought the hardware at https://www.hub360.com.ng, here in Nigeria to use ARDUINO 1.6.5, and I uninstalled the 1.8.4 and installed 1.6.5, and when I tried I got an OK from my Serial Monitor.
I have resolved the problem I am having and I say "thank you so much dannable and lastchancename, you guys are a blessing to this forum and because of your speedy response I will like to stick around this community much longer than I thought, I love you guys".