The Kilz is a good primer, and its critical that you apply a good even coat. Brush the edges and trims if they are panel doors and roll the flat areas. Ideally a power sprayer makes quick work of this if you have the place in which to set out the doors to work.
I find it much easier and more attractive to take all the doors off their hinges to paint them. You can end up with a more even coat if you take off the doors and lay them down on sawhorses or something to paint. It's easier to get the edges this way, and you don't have to worry as much about the hinges.
All good advice so far.
Going from stain to paint has common pitfalls because the varnish is so much harder than paint. If not done correctly, this can easily lead to future peeling or chipping, so the prep work is paramount.
Lightly sand the doors smooth and wipe with a deglosser; this will remove any oil or polish that commonly occurs, especially around the handles.
'CoverStain' covers a little better than Kilz and is available everywhere that you find Kilz.
Once sanded, cleaned and primed, you will need, at minimum, two coats of finish paint to cover the dark surface.
Spraying is typically done best by pros. The right 'mini' roller with the right paint can achieve similar satisfactory results.
Thanks so much for the advice. I feel more confident in receiving good results.
We had a huge amount of doors and trim to paint in our house...currently doors are done and just have second floor trim to finish. We painted them by taking them off door jams and spraying them. I agree the prep work is of the advice above... covering hinges with painters tape, putting two screws into bottom and top of door about a foot out from edges. We rested the doors on cinder blocks using the screws so as to not smear the paint.
Tried and true... Before you paint. Remove the door, hinges and all hardware. Lay the door flat and if you have a stain on them then you need to apply a deglosser. You can buy a small can from HomeD and fine steel wool. It's not physically demanding (like buff shining a car) but you do want to make sure you scrub the entire door lightly. Let it dry thoroughly and then use either a lightly damp cloth or a tack cloth to remove any bits of steel wool or deglosser residue. At this point if you have to fix any dings or dents then do so and sand the area down. Wipe down and let dry. Be sure that you sand in the direction of the grain of the wood. Apply Kilz and then your color. Be sure you buy the right Kilz There is one made just for this work and another that is more a mold and mildew remediant. Good luck. I just did 24 doors last year.