Proof for the safe way to climb a mountain without getting altitude sickness?

I am terribly afraid of mountain sickness and my mom even claims that I've inherited it from the family. I've heard some family members get it during their trip to Lhasa, which ruined their trip at that point.

Yet I'm planning on travelling to Switzerland to see those tall mountains, and it would really be a waste if I did not visit some famous ones NEAR the summit, such as take a hike on the Matterhorn montains.

The fact is I know that you could easily get mountain sickness if you 'ascend rapidly', but there is no real definition. It's very abstract and I don't even know if ascending by cable car is 'rapid'.

However I've heard three things, which I am asking the proof or evidence for :

1. Is ascending by train 'ascending rapidly'? What about for cable car then?

2. I've heard that if my visit to the high altitude places is less than 6 hours, mountain sickness would not occur that likely, unless I did vigorous exercise. Is that true?

3. People say we should rest before ascending further to prevent mountain sickness. How long should this 'rest' be?

The answers to these questions would greatly determine whether I visit some landmarks of Switzerland, and I hope someone could offer helpful suggestions. Thank you very much!