Sounds like a nice project you have planned. I've just refinished a set of Heresey 1 in Mahogany but have always wanted Cornwalls. I'm going to risk sounding long winded, but I've notice a few people have asked for a detailed description of the veneering process.
As a first project, applying veneer is not really technically difficult, but you must be meticulous and plan it out. Although I prefer the good old fashioned contact cement method (old habits die hard), iron on veneer and heat activated poly vinal acetate (white glue) work just as well, need fewer tools and make less of a mess. I used to work in a cabinet shop which did a lot of veneer work and we would always strip cabinets of finish before we applied the veneer. You never know how the contact cement or the heat from iron on veneer will react with the old finish.
Fill any scratches or busted corners, sand, and plan your veneer layout. Ensure the veneer's grain flows unbroken across the tops and sides (like a waterfall). Cut your pieces 1/4" larger on all sides. Start with the bottom, then the sides, next the top, and last but not least the front. Apply contact cement to both the cabinet and the veneer using a brush and roller (thin the cement with laquer thinner if required). Apply a second coat of contact cement to the edges and corners of the cabinet and areas where the 1st coat might have soaked in too much. Wait for the glue to dry to the touch and position the veneer on the case very carefully.
Start at one edge (you did cut your pieces square didn't you?) and press on firmly (using a wood block if you don't have a roller). An old trick is to use a hot iron (use your wife's best one) to help really kick off the contact cement on the edges of the case and on any longitudinal seams in the veneer. Trim almost flush with a really sharp razor knife (use a new blade) and a really sharp pair of scissors (sharpen or buy a new pair) and sand flush using 150 grit sandpaper around a wood block (finish off sanding with the grain to avoid scratches that will show in the finish), or a quarter sheet orbital sander if you have one.
Continue the process until all the sides are done. Use iron on banding for the fronts and miter the corners using your really sharp sissors or. Sand smooth with 150, then 220 sandpaper and lay on whatever finish you desire whether it be oil, polyurethane etc.
Hopefully this helps explain the process. Good luck.