You'll not regret getting rid of the old carpets (except about half-way through doing it!)
Get a strong razor knife with plenty of spare blades and cut around walls, furniture, etc. The carpet is likely held down with a bazillion rusty little staples, so you'll do a lot of tearing and swearing. A dust mask is a good idea because you will be stirring up a lot of nasty stuff from underneath that carpet.
We used a "floating" click-lock flooring (vinyl planks)from Amstrong rather than something with adhesive. Easy and quick to put down, looks like wood, warm and fabulous.