One of the few Nintendo franchises that I haven't played yet. What's the best entry to start with?
Megamix on the 3DS gathers most of the minigames from the entire franchise (with some notable exceptions) plus some new ones. It has some QoL improvements like being able to see how accurate your inputs are and supports both button controls and touchscreen controls. It also has offline multiplayer if ypu are interested. All in all, is the most complete entry. Great for a beginner (despite its silly story. Yes, this entry has a "story", but don't expect much in that regard)
However, I strongly recommend you to check Rhythm Heaven for the NDS and/or Rhythm Heaven Fever for the Wii before you jump on Megamix. Not only they have some exclusive minigames, but they are much challenging games in general. Megamix is a cake walk to finish because you can beat minigames even if you miss a decent amount of inputs. You have to mess up real bad to lose... And that doesn't feel right. The DS and Wii version are more "fair" in that regard.
Also, the NDS entry in particular has a more original control scheme. On Megamix (3DS) the touch controls are limited to tapping the screen. On the DS however, minigames use different touch mechanics like sliding, which in my opinion feels more varied and fun.