Wiring an Accessory with a Relay

Dale I would prefer to use an insulated rail but to be honest with you I had a hard time figuring out the wiring connections provided to me from other forum members - I'm really bad at understanding electrical connections - everything needs to be in "simple" terms (i.e. the red wire goes to this, the white wire goes....).  

I have 2 MTH crossing gates (red, black, white, yellow and blue wires).  I have a double track between them with trains going in both directions.  

My LHS supplied me with a relay.

Paul

Dale - an "easier diagram"  would be very helpful - thanx so much!  The pic you sent me has a "rectifier" - that's something my LHS did not give me.  If you could do that for me I would greatly appreciate it - here is what I'm working with:

2 tracks with trains going both directions on both tracks

2 MTH crossing gates (#30-11012)

10-12 VAC DPDT relay & 8 pin mini socket (made by NTE electronics)

If your e mail address is in your profile I'll send you my snail mail address.

Thank you, thank you, thank you, thank you!!!

Paul

Dale - you're most gracious - thank you for your much needed help!  I am in absolutely no rush so please take as much time as you need.

Regards

Paul

Dale - this looks easy to follow.  I know it took lots of your time but I do appreciate it.  Your explanations simplify things for me - I'm sure when I do this the next one will be no problem.  

Thanx so much

Paul

THANX FOR THE ENHANCED PIC, DALE - 

Great information here and thanks to all the posters. I have a question about the function of the blue wire in the MTH 11012.  Dale indicates to leave it alone in one of the circuit diagrams.  I had this sketch below from azatrax, and seems they think the blue wire completes a loop of some sort.  I can't find any internal wiring diagrams from MTH.  Would one of you experts help me out on what the function is for the blue wire and why this difference in views? Thanks.

My LHS also feels that the blue wire should not be connected to anything.

Paul

Guys, I hate to ask but.... what is the purpose of the blue wire on the MTH 11012?  I was looking at the excellent sketches from Dale and ran across this below from azatrax.  Azatrax uses the blue wire apparently to complete a circuit (can't find any internal wiring diagrams for the 11012).  In the recommendation from Dale, it says blue is not used?  What does this wire do?