When it comes to home improvement, there aren't too many jobs worse than stripping paint! I can offer you a few paint stripping tips to make it a bit easier but with several doors and trim having multiple layers of paint, you should prepare yourself for a long project!
First, presuming you want to remove all the paint, you have three options: sanding, chemical stripping and heat gun stripping. Before you decide which paint stripping method is best, you should first make sure the doors and trim are worth stripping to raw wood. You might want to work on one in an inconspicuous area to get a sense as to what kind of wood you might be discovering after all that hard work.
Your friend is right to suggest the heat gun. A steamer is not the right tool for paint stripping, although it works nicely to remove old wallpaper. However, with a heat gun you need to be very careful for several reasons. First, a heat gun can be a fire hazard. Second, it can be a health hazard. Stripping all that old paint will produce some very toxic fumes. When stripping paint, you'll need to work in a well ventilated space and wear a very good quality respirator (NOT just a dust mask ).
As for chemical strippers, I have had good success with Rock Miracle. Again, you need to wear appropriate eye, hands, face and respiratory protection.
Sanding is almost guaranteed to be a part of your paint stripping project too. Even if the heat gun or chemical stripping get off a majority of the paint, you'll need to sand to remove the remaining bit.
Also, since this is an old house, you may very well be stripping off lead paint, which can be a health hazard to you as well as your children. You would be wise to have the paint tested before you begin this paint stripping project to make sure you are not creating an unsafe condition.
Finally, by the time your paint stripping project is at an end and you have stripped such a large amount of painted wood, you might find you'll be better off just removing and replacing the trim and perhaps even the doors. I have an 1886 home and did this in almost the entire house. As a result, I was able to replace the trim to look almost exactly like the original but avoided hundreds of hours of labor in the process.