There are a few issues here, I'll try to cover all that I think should be addressed.
Rather than address what to buy first, let me cover the bull-barrel question: Chances are, you won't need it right off the bat. The 10/22 is a nice little gun, but the bull-barrel is somewhat specialized. Until you can start out-shooting the inherent accuracy of the base rifle, or shoot so much that you need the extra barrel area to aid in cooling, it's really not necessary. Of course, I spend a lot of time minimizing the "necessity" argument when talking to the wife about something I really want.
As to which rifle to get first. . .well, it sounds like a cop-out, but. . .that's all up to you. It comes down to what you think you need or want. The 10/22 is a great gun to learn shooting fundamentals. It can also provide meat for the pot, get rid of annoying pests, etc. It's also quite fun and comparatively inexpensive in both initial investment and continued purchase of ammunition.
The AR is more of a specialized rifle. While I'm not wanting this thread to deteriorate into a this-vs-that discussion. . .an AR is a good defensive rifle or an excellent rifle for the various shooting sports such as 3-gun. With proper ammo selection, it is an excellent post-TEOTWAWKI General Purpose Rifle. (oh, and they're a lot of fun to shoot)
But, compared to the 10/22, you're going to pay more for everything (the base rifle, any additions and ammunition) except, perhaps, for magazines. Replacement parts are easier to get and can usually be swapped out with minimal difficulty.
However, one caveat: don't buy your 10/22 from a box store. Spend a few more bucks and get one from a proper gun dealer. There IS a difference (or at least there was 5 years ago). Ruger has different specs for guns bought in very large quantities to be sold at larger non-specialized stores. There's a different metallurgy for the receivers and barrels, the finish isn't as clean and the fittings not as tight.
Chances are, you won't notice them very much unless you do decide to go for the super-accuracy or high-round count mindset.
If my son were to ask me the same question, though, I'd suggest he buy the 10/22 from a reputable gun dealer, learn to shoot it and then go after the AR.
Hope it helps.