In addition to what others have posted (of which I agree) here's a rule thumb when selecting a palette: 60-30-10. 60% would be your prominant color, 30% your accent color and 10% your trim color. This would apply to both the interior or exterior.
To save time and money and to provide continuity within your space, you could paint all spaces in the same palette, use all the same brushes and rollers (if done in a timely manner) and buy 5 gallon bucket(s). That way you're not changing out or cleaning rollers to change to the next color for another room and incurring additional expense and time.
Hopefully you can stage each room with diversity in color to compliment your paint color choices.
Trust your local paint supplier to have the most current color selection and don't hesitate to ask their guidance in selecting your neutral color pallette. There are some beautiful ones out there! Neutral doesn't have to be blah....
You're on the right track! Best of luck to you.