Let's not scare people unnecessarily. Thousands and thousands of people visit the Rocky Mountains from all over the country year round to hike and ski and most do not have a problem. No sense making it sound worse than it is. The tourist and ski industry with some of the best skiing in the country, would suffer greatly if people were dropping right and left from altitude sickness when they flew in to ski or hike.
It is definitely a thing and you do have to be aware and smart, but the more serious altitude sicknesses are not all that common. Drink lots of water, refrain from copious amounts of alcohol, and ascend over a few days just as you are planning.
Acute altitude sickness (the mildest) does not usually occur until you ascend to 8,000 ft or more. It is rare at Denver's 5,000 ft. That is why the OP's plan of spending a couple of days in Denver first is exactly what is recommended.
Over the past 40+ years, we have had numerous relatives from below sea level Europe visit, friends from the East Coast and other flatlanders, and have met many others through other friends, and have yet to meet anyone who experienced more than a headache for a day and that was only a very few. If you do experience coughing or shortness of breath when up in the mountains, get to the ER and get off the mountain asap (Flight for Life will take you down.) But you don't hear of it very often in the news of tourists experiencing the more serious forms of altitude sickness.
But it is fun to let people think they are going to suffer. At Mile High, this is what the visiting team sees as they exit the visiting locker room. It is on the wall right opposite the door:
Just be aware but don't be afraid.
Here is a great article:
https://www.bouldermedicalcenter.com/understanding-preventing-altitude-sickness/