The first step is going to be to prime the chipped areas with a bonding primer so that whatever filler you use can adhere to what looks like stain/varnish underneath. Unless you are going to apply an oil base top coat, all the trim will need to be primed with a bonding primer to ensure adhesion of a water base top coat.
You will need to carefully remove the loose pieces around the chipped spots. Misting around the spots with water, then gently prying off the loose material with a putty knife would be one way to do it. Having the area slightly damp before you scrape on it reduces the chance of airborne lead dust. Let dry completely before priming.
As far as filling the chips, one way would be to use a spackle product. Dap Crack Shot is a good spackle. It dries fairly fast, is easy to sand, and doesn't flash paint too bad.
Filling the chipped areas with latex caulk is another option. Caulk will adhere better than spackle, and help prevent future chipping around those areas. The draw back to using caulk is that its harder to work with and it can't really be sanded. You would just have to apply the caulk as smoothly as you could and hope for the best.
Using caulk would not likely make the chips totally disappear, but it would be more stable and hold up better long term. Maybe you could use a little caulk to initially fill the areas. Kind of glue the edges of the remaining paint down with it, then fill with spackle on top of that so you could sand it flush.