Fun!
We did this years ago. When the kids got too old to ride in the Burley trailer we switched them to the back of a tandem. We bought an entry level (but still nice) Burley Duet road tandem. Burley offered a child adapter kit which we installed on the back until the legs grew longer (this never happened as by that age they both wanted to ride their single bikes). The various adapter kits for growing kids work quite well.
After the kids were grown and gone we started riding the tandem (without the adapter kit) when I wanted to start cycling again and when we found out how much fun it was we upgraded and got a nicer tandem.
If I were you I'd find a gently used entry-level road tandem with an adapter kit and get going! You'll need to be sure the front fits you well, but the rest should be pretty adjustable for the child. We found that if we turned the stoker bars over (so the "horn" is facing towards the stoker instead of the captain) it makes them easier to reach, but then we started our kids on the tandem at age 5 so maybe you won't need to do that. It seems like gently used tandems are pretty easy to find because people tend to get these grand ideas and then for one reason or another they park their tandem in the garage and don't use it. Our sat for 10 years and we just sold it two months ago. It was a $2200 bike new, and we sold it for $800 and it was in mint condition.
Another entry level road tandem that I'd recommend is the Grand Junction model by daVinci (www.davincitandems.com). Do use toe cages on the pedals, even if you choose the kind without straps (for the stoker).
For transport we bought a Yakima tandem rack for the top of our car. I personally think it's difficult to hoist a tandem up that high to put in the rack but then I wasn't the one who had to do it.
Have fun ! You'll probably find this is the best thing you ever did!
2010 Specialized Sirrus Comp (XS)
2010 daVinci Global Venture