Hi Sid Unfortunately, this is an all too common problem. What typically happens is that homeowners do a "fast" spruce up of a house about to go onto the market, that entails putting paint on poorly prepared surfaces. Then in a year or so, the new owners has to deal with peeling paint.
Regretfully, you will have to remove all new paint. Putting excellent paint over a paint that is poorly adhered will not improve the original paints adhereance...it has to come off. If you are lucky, they did a terrible job of preparation and the new paint will scrape off easily, using a good sharp scraper. Don't use chemical strippers or heat to remove the newer paint...you'll have a terrible mess if you do. If you run into paint that is well stuck on, just taper the edge of the paint with coarse and then fine sandpaper until you can't feel the paint edge with your fingers. Then when down to the old and well adhered paint, do what the former owners should have done; using a random orbital sander and non-loading sanding discs, break the glaze on the old oil paint (it will dull the surface considerably) then wash off the sanded surface with a 50% TSP solution, rinse twice with clear water and allow to dry thoroughly. You should be able to apply your finish latex coats directly to this prepared surface.
If your door/window casings have many years coats of paint, it may be easier to replace them. For doors, you have a couple of options. You can take them to a professional stripper and have them dipped, which will strip them back to bare wood...but you will have to detail them. Or you can scrape/sand them down by hand, taking great care around the rounded edges of the rails and styles...if they exist.
This will be a time intensive labor of love, but if you take the time, be patient and do good work, you'll love the results.