Measure your walls and make a plan drawing of your wainscoting project. Note locations of fixtures, electrical outlets, etc. Use a level to check your corners for plumb. If not plumb, mark plumb lines on the wall to use as reference points.
Note: Take this time to plan out where you want the ends of the plywood to fall on the wall. On this project, we are placing our stiles every 4', which means our plywood butt joints must fall at 4' or 8' increments. The stile is the vertical part of the trim that covers the butt joints of the plywood.
Mark the walls with your top-level line. This line will represent the top of your 3/4 ply. Find the highest point on the floor along the walls. From this point, measure up 1/4" higher than the height of your ply. This will allow a gap between the bottom of the ply and the floor. The oak bottom rail will eventually cover this gap.
From this mark, use a chalk line with a water level or a 6' level to draw a line all away around the room.
Mark where the ends of the plywood (butt joints) will fall on this line.
Begin from an inside corner. With help, hold the sheet of plywood up to the level line and using the 3" wood screws, screw the 3/4" ply into the studs all away around the edge of the ply. Place the screws about 1" to 2" in from the edge. In addition to the screws around the edge of the ply, secure the middle of the 8' sheet of ply (every 4') with a few screws as well. These screw heads will eventually be covered by the oak rails and stiles.
To make driving in the screws easier, pre-drill first.
Continue this method of hanging the 3/4 ply until all is installed.
Note: Pre-rip all of the trim (rails, stiles and cap) to the proper width. Cut the trim to length as it's installed. Also take this time to sand all of the trim.
First install the bottom rail. Our bottom rail for this project is 6" wide, ripped down from the 1 x 8. Start in an inside corner and work your way out leveling as you go. Maintain the 1/4" gap from the floor, staying level with the bottom of the ply. This gap will be covered by the 34" oak 1/4 round.
Next, install the top rail. For this project 1" x 4" is being used for the top rail. A 1" x 4" is actually 3-1/2" wide, so we are using this as is. Start installing the top rail from the same inside corner you started the bottom rail from. Keep the top of the top rail level with the top of the ply. Continue until the entire top rail is installed.
Use 45-degree miters for your butt joints when joining the end of one rail to another. This technique gives a cleaner look than a typical 90-degree butt joint.
Our stiles are 5" wide, ripped down from a 1 x 6. Again, start your stile installation from the same inside corner and work your way out every 4'.
Remember, you have already pre-determined where your stiles would fall. Cut your stiles to length as you go. The length of each stile may change slightly.
The cap sits on top and covers the edges of the ply and top rail. Our cap is 2" wide, ripped down from a 1" x 4". This will leave a 1/2"overhang. Starting from the same inside corner, install the cap. Use 45-degree miter joints at both inside and outside corners.
Finally, install the 3/4" 1/4 round. Starting at the same inside corner, work your way out. Again, use 45-degree miter joints at both inside and outside corners. Unlike the bottom rail, the 1/4 round will sit on the floor and hid the 1/4" gap.
Touch up any gaps in your joints with some oak wood putty and let dry. After some final sanding, you're ready to stain.
Using a pneumatic 18-gauge finish nail gun will dramatically speed up the installation of your trim. It will also save any unnecessary damage to the wood that a hammer might otherwise inflict.