Veneers can be tricky. When I get to work on them I always give myself extra time. Make sure it's very, very clean. I usually clean with soap and water and if the piece is over twenty years old I will use alcohol to get the last bits of oil residue off.
Sand lightly. And glue the parts of the veneer that are lifting.
I will use a bonding primer first. Let it cure overnight. The next day take your fingernail and see if it scratches up the primer. If it does you will need to sand a little more. then prime again. and let sit for a couple of days. I always get over exuberant at this point and start painting before my primer is cured hard.
With veneers I always paint an under color first and let dry. If you sand them too much you will sand right through the veneer. Not Good. and not easily fixed.
I paint my color of choice and instead of sanding the edges back for that "old" look, while the top coat is still damp I will take a damp cloth and use it on the edges and handle-holds. Wherever a hundred years of handling would be seen. This keeps you from taking the chance of hurting the under veneer. Seal it good when dry. I use a hard coat clear matte sealer.
Most of all have fun.
You actually have several options. Regular interior latex paint will work well if you use a good bonding primer first. I have painted a lot of furniture, including veneered and laminate furniture, and latex is my preferred paint. (Endless color choices, several choices of sheen level, reasonably priced, the color is consistent and it produces a nice, smooth finish.) You can sand the edges to distress it after the paint has cured. If you sand before it has cured well, latex has a tendency to peel or "roll" up in little balls, which is definitely not a good thing. (Don't ask me how I know that.)
Chalk paint is VERY popular right now because it will stick to just about anything without any prep work. However, you should use some kind of protective coat on it (wax or poly) to give the matte finish some luster and protect it from wearing/chipping off. There are several commercially made brands, or you can make your own. Here's a link for a blogger who compared the different homemade chalk paint recipes. http://salvagedinspirations.com/best-homemade-chalk-paint-recipes/
Another option is milk paint. I've tried Miss Mustard Seed's milk paint products and it is good if you like the more rustic/imperfect finish it gives. Milk paint basically requires no prep either but it can be a little unpredictable. Some surfaces can resist taking the paint and it may chip off in places. If you like and want the chippy look--great! If you don't want chipping or flaking, Miss Mustard Seed also has a Bonding Agent that can be added to the paint so it sticks better. Like chalk paint, milk paint dries with a matte finish and should be coated with wax or poly to give the paint luster and protect it. I really like Miss Mustard Seed's wax. It is soft and creamy, and super easy to buff off.
Ditto word for word what @Shari said.
@Karen I use white spray shellac, giving a light mist-spray coating, (lightly mist..don't try to cover with one heavy spray , but make sure item covered).... on most all my redo old dark pieces.. when dry NOTHING bleeds through, you can now paint to your hearts content!.. this is the greatest..purchase Lowes & Sherwin-Williams
thank you everyone for taking the time to answer my questions!! I'm of course going to try what I think sounds like it's the easiest way! I'm anxious to get outside and follow Jan B.'s process! Now if this darn MN snow would go away...already!!!