The doors of this gate are much like a cabinet door. They are built using a "frame and panel" design aka "rail and style" method. If I were to build these gate doors I would finish nail them with 15 ga or 16 ga stainless steel fasteners and glue them together with a polyurethane/waterproof glue, instead of leaving the panels floating. Or I would at least glue the rails and styles and finish nail the panels. It all depends on what type of climate you live in. That along with properly attached heavy hinges will allow for many years of use before the doors begin to sag under their own weight. When that time comes string a cable and turnbuckle from the high hinged corner to the low unhinged corner. Tightening the turnbuckle will lift the sagging corners back up.
Your doors contain a "muntin," it is the stiffener across the middle of the door that will help it to stay rigid as well.
If you are unfamiliar with frame and panel construction and rails and styles read this attached wiki article... it does a good job of explaining the basics.
Eh?
http://en.wikipedia.org/wiki/Frame_and_panel
You need to cut a channel (called a dado cut) into the rails and the muntin. Note the groove where the panel sits in the frame and panel article. This cut is usually made with a dado blade in a table saw, or a 3/4" dado router bit. The ends of your slats will fit into the dado cut. The dado cut will be on the boards that run horizontally for this project, again the rails and the muntin. Do not nail your slat boards to the back of the frame! Your slats are considered the panel in this type of design.
You should also read up on mortise and tenon joints. The joint that holds the rails and stiles together is much like a bridle joint (aka butterfly joint) only it does not go all the way through the wood stock.
Glue your mortise and tenon joints with gorilla glue. Finish nail your slats, top and bottom, with at least 2 nails each. It is your choice if you want to use glue on the ends of the slats as well. Use ratchet straps (the type used to hold down a load in the back of a truck) to hold the door together while the glue is setting up.
If you live in a rainy climate your gate will sag faster than a dry climate. Support the gate with a cable a couple of years down the road. If you are in an area of Canada where there is a dramatic temperature difference between seasons and not a lot of rain, don't worry about the support cable. Don't forget to drill drain holes, 1/2" diameter between the slats on the muntin and bottom rail. This will prevent water from sitting in your dado cut. You could also glue 3/4" stock between the vertical slats to reduce the space where water can sit.
I hope this addition is not too confusing. I would be happy to clarify anything else through e-mail (see my profile), not edits to my original post.
tools you will need to make these gates...
Table saw OR router mounted in a router table with bit, if using a router make multiple passes as removing too much stock at once is dangerous and ineffective. This will be for the dado and tenon pieces.
Power drill with bits, a SHARP 3/4" wide chisel and a hammer to make the mortise.
A combination square to draw your joinery layout on your lumber.
http://en.wikipedia.org/wiki/Mortise_and_tenon
http://en.wikipedia.org/wiki/Bridle_joint
http://www.guttersupply.com/p-gutter-tools-combo-square.gstml