Hi Kim Building screens is a little trickier than you might first think....although with some planning and the right tools its not too difficult.
You don't mention size. The larger the screen area, the more care that must be taken to make sure the finished screen is tight and without wrinkles.
You must start with a solid frame, as the tension in the screen can amount to a lot of pull that can bow the frame material if it isn't large enough. For a typical 27" X 58" double hung window, I use 5/4 straight grain yellow pine with a moisture content of 8-10%....the lower this reading, the better. I'm starting to use finger jointed and pre-primed pine, as it is very flat (not bowed) and I've found the moisture content to be acceptable. I use only through mortise and tenon joints, although bisquits with exterior grade glue or full lap joints will prbably be ok. The M&T joint will help ensure years of use. The traditional method of holding the screen is to cut 3/4" wide by 1/4 to 3/8" deep rabbets around the inside edge of the frame, stretch the screen and staple into the rabbet, then put screen mold over that. Another method is to route a 1/4" dado around the inside of the frame about 1" in from the opening, lay the screen over the dado on one end and tap a 3/16 dowell down into the dado. You'll need to use a metal screen to ensure a tight fit (I don't think that vinyl is thick enough to make the dowell tight in the dado. Then stretch the screen and tap in the other dowells. This is definitely a two person job.
Be sure the wood is at least primed before putting in the screen. I like to put on the finish paint before installing the screen material, taking care to install on a padded surface to avoid scratches. Nothing looks tackier than paint dripped on a screen **smile**
Hope this throws some light on what you are wanting to do.