The key to a successful furniture painting project is in the preparation. Surfaces to be painted need to have a dull finish so that primer will adhere to it. TSP, TSP Substitute, or liquid sandpaper can be used to de-gloss and prep surfaces, but if you have a very glossy finish, as in the table example below, liquid sandpaper is the best product because it is great alternative to sanding for speed and dust control.
Using the liquid sandpaper is my preferred method of furniture prep when the finish is intact. I don't like sanding because of the dust that it creates. However, you should take precautions when using the chemicals by reading the directions. If you have flaking surfaces, you will have sand the surface. All flaking surfaces have to be removed. If you have veneer damage, simply remove the loose pieces and fill the voids with a wood-filler and sand until smooth. My preferred primer for small pieces is the Kilz Original oil-based spray primer. This makes for quick work, but you must have a well-ventilated space and a spray area to catch the over-spray. If you don't have that, Kilz Premium water-based liquid primer is the next best thing.
If after you prime with water base primer you have bleed through of brown you may have to spot prime with an oil based primer because that brown may bleed through to the finish coat. I believe it is shellac in the old finish that is bleeding through. Use 4" high-density foam rollers for a brush-free finish. I use a brush for the edges and corners first, then use the roller for overall surface painting. The trick for no drips is to , always brush "off" of the edges.
For the best top coat use quality paint, such as Benjamin Moore. Use eggshell finish for a washable matte finish. Pearl finish is a step above eggshell and semi-gloss is the glossiest you can get in latex. Choose your finish based on the look and function of your piece. Typically, you will need one prime coat and two top coats. If you are distressing your finished piece, you may only need one top-coat. Have fun transforming your pieces!