I sort of glossed over the point about the possibility of a nailing strip on your existing door frame. This would prevent the frame from pushing out as I described. You may only notice that there is one when you try to push the frame out, as it would be concealed under the siding or whatever you have on the outside of your house. In this case, you would not want to push the whole frame out as it would break or dislodge the outside finish, making even more work for you. Generally, I would in this case simply take the reciprocating saw and cut the frame into sections, then use a crowbar to pry the sections away from the wall frame. The nails that were driven through the flange will remain, but the flange itself, usually made of plastic or aluminum, should break easily. After the frame is removed, try to remove any excess flange that broke off and is still attached to the wall frame, and drive the flange nails as flush as possible.
Replacing a door can be tricky, but not dangerous. Assuming everything was built properly, this project should not cause the house to fall down. The biggest problems you will face will be getting the new door aligned properly so that it closes and seals right, taking care of the soft floor (whatever that turns out to be), and completing the ascetics of the inside and outside so the project looks finished. Of course, every project has it’s hidden problems waiting to be discovered, but you simply take them as they arise. Just take your time and don’t be afraid of admitting when you get in over your head, and chances are something won’t happen that’s irreversible. In fact, if you determine professional help is in order, each step you do yourself will make the completion less expensive.
If you want, when you have removed everything you can post the results, and maybe someone can give you advice on what you find, and also with the selection/installation questions on the new door. Remember, none of us were born with all the knowledge…we had to start out by asking questions just like you are doing.