What are some steps an undergrad can take to get on a Ph.D. track?
Many students who want to study for the PhD in the US apply directly at graduation from their undergraduate education. A MS isn't needed for most fields, but might include one along the way for little or no additional effort. But expect that such a doctoral program will include additional coursework in your field prior to starting research, and, perhaps, prior to choosing a research advisor. 

The advice you were given is good, but the MS isn't actually needed (most fields). What you need is a solid record of accomplishment and good letters of recommendation that suggest you will be a success in graduate study. 

The advice about going to more than one school is so that you have more opportunities to view a variety of approaches and a wider range of ideas than you are likely to get if you stay at the same institution throughout. 

In some doctoral programs you can get a MS just by asking for it and paying a fee (some paperwork). In some you write a thesis of lesser scope than a doctoral dissertation, but possibly giving some practice with research and writing. 

The path might be something like 6-8 years, possibly as long as 10. 

You need a way to support yourself. The common way is to serve as a TA in your doctoral department. Research Assistants are open as well, but to fewer people in some fields. 

But in the US, the best advice for right now is to apply to a few programs that interest you, including some that are more likely to take you than some others.