What undergraduate degree should I take to get into data science?
If you can manage it, I suggest focusing more on Statistics. Statistics can be a demanding field, and takes a lot of effort to attain a degree. Of course, a Data Scientist also needs several skills besides Statistics, such as Software, Data Engineering and Visualization, but Statistics knowledge/skills are the most rare to find, and therefore can be the most valuable.
I worked the last 8 years in a data science department and found that when looking to hire a data scientist, it was often difficult to find someone with a solid statistical background. Most candidates had software, engineering, data and visualization skills. However, someone with a stronger statistics background would be hired over the other applicants, and usually with a higher salary.
Internships are very valuable, they will help you get your foot in the door before you graduate. Some companies may even offer you a job during your internship if they like your work. Some of the bigger companies will pay for your continuing education as they will get a better employee. It is a good investment for both you and them.
A Master's degree will also be beneficial, but may limit your chances to get a lower paying entry level job. Larger companies will usually prefer a Master's over a Bachelor's degree.
Response to your followup question:

I strongly suggest doing the internships, they can dramatically
improve your chances to get a job when you graduate, especially with
the companies you did the internship with.
The choice between Masters with little experience vs Bachelors with
several internships would be much more difficult to make.  Overall,
the better choice would be to get the Masters degree with a few
internships along the way.
Also, I suggest going with Statistics all the way, Bachelors and
Masters. Several of the topics in Data Science will evolve over the
6+ years it takes to get the Masters, but the underlying Statistics
will change very little.
Maybe get a minor degree in Data Science or a Statistics degree with
an emphasis in Data Science.
You can also consider looking for a job in between the Bachelors
degree and Masters, and if you find a good job, you can always get
your Masters degree while you are working and earning money.