This is actually something I am currently pursuing, and I believe that the best way is to get a full CS degree first. I got a normal CS degree at Carnegie Mellon. After graduation I felt that while I knew lots about coding and had a strong base in C and systems, I was still lacking in many areas I would need in the industry, like more advanced graphics, AI, and general game engine programming. I thus applied to the masters program at DigiPen, and am now about a month in. So far this is exactly what I was looking for, and I am rapidly building a strong foundation in making games. However, I am very glad that I got a normal CS education before coming here. I do not know how it works in RTIS (real time interactive simulation, the main undergrad degree for programming games at DigiPen), but I know that if I ever need to code anything, I have the ability to. I also know that if for some reason I decide I want to leave the games industry, I have the skills I need to pursue many other jobs.
DigiPen is wonderful for undergrad if you know you will want to work in the industry, but as I said, I do not know how broad the CS material you are introduced to is, so if you decide to go somewhere else it might not be too helpful. As for the other game programs, you really need to do your research. In researching for myself I found that most of the programs are worthless, even inside of the game industry. I seem to recall finding a few others that seemed to be worth something in undergrad, but DigiPen was the only masters program I found that seems to actually have value once you graduate.
Seeing as I go there, I should probably just say that I am biased towards DigiPen, but I felt this way before I got in as well.