Asbestos containing materials are found in a number of building materials manufactured from the 1930s to the early 1980s. It is a mineral fiber with exceptional insulating, heat resistance, and sheer strength. Asbestos is not toxic. It is a carcinogin that may increase risks of certain cancers over a long period of time (30+years); but exposure is not by any means a certain predictor of health problems. Think of it more like smoking. A person can have considerable direct exposure without any ill effects for a lifetime; while some sensitive individuals seem to be affected by indirect exposure. Abestosis is strictly an occupational disease rarely seen any more. You would need to be in daily contact.
Asbestos fiber is natural and in fact is generated from the serpentine rock formations during constuction in my area. Zero lifetime exposure is not possible even if you never work or live with it in your home, and everyone has been exposed to hundreds or thousands of fibers. Our objective is to minimize such exposures, particularly to populations at greater risk...children.
A typical home survey will include samples of any suspect materials. Could include wall materials, joint compound, flooring, mastic, thermal insulations, and cement siding. The only potential risk to you are materials you will specifically disturb. So a sample of flooring might be wise to take before you make a decision of the best approach to remove or manage the material in place. The same is true of furnace insulations, siding and other materials.
Your concerns about the materials in this older home are understandable, and the value of your peace of mind is really what to weight against the cost of a qualified inspector. Most of the materials you can sample yourself and deliver to an analytical lab for much lower costs, but recognition is the key here and that is the expertise you would hire. That said, I would encourage you to stay away from individuals that speak in tones that tend to panic rather than educate.