I would remove the door trims as well and just retrim in a beautiful moulding that you love.

Should the door trim match the baseboard trims (size, pattern?) or are they different? Sorry for my lack of knowledge in this area.

No, installations don't always match exactly in the way the detailing of the boards is cut, although they should be of the same general style (traditional, contemporary, etc.). There are actually different pieces for different purposes -- baseboard trim, chair rail trim, crown moulding trim, door and window trim. There are a variety of styles to choose from. HD or Lowes probably stocks the most popular types without special order. You'll probably find them in sets as far as styling goes.

Pictures, pictures, pictures will show you how things are put together and just how much it all varies. Look to see what type of detailing you like and how wide of a trim appeals to you. Of course, if you like really wide trim, it may not look right in your particular space! So you have to take all things into consideration.

The contemporary trims are usually very simple, flat pieces. At 1-1/2 inches it sounds like you might want to go wider.

If you remove the trim around doors, do you also need to replace the door jams?