My wife and I compete in Rimfire Challenge. We practice at least weekly, and attend a monthly match, plus state and national matches annually.
I am also an Appleseed fan (got my Rifleman after a few tries), we've both done their Pistol Clinic as well, and I do a little Precision Rimfire on my own.
To be honest, the ideal equipment for these events, for example, are rather different!
Yes, you could use the same rifle at all three ... and if you can only have one, that may be what you need to do. But ideal, it's not.
Rimfire Challenge is rifle-and-pistol, by the way, and you'll need five magazines for each. Accuracy and grouping isn't a primary goal, once you realize that the targets are large! But reliability IS a big deal. Your firearm cannot have misfeeds and so forth at rimfire challenge events if you want to have fun and be at all competitive. You'll also use a lot more ammo in rimfire challenge than in other disciplines.
Appleseed, you'll want two magazines (loads of 8 and 2) and a spare. Appleseed teaches you solid riflery skills, including correct use of a sling and the ability to shoot from standing, seated, and prone, with both fast and slow fire, and with mag changes. The rifle itself only needs to be 4MOA or so accuracy. Project Appleseed events are a lot more about the indian than about the arrow, to use the expression. And you leave with a deeper sense of what that part of our national heritage is really about, and what part of it is you, too. The only competition is versus yourself.
Hope some of these comments are relevant and helpful.