Your basic choices are a child-back tandem as discussed, or adapting an adult one. Adaptations come mainly in two forms.
Kiddy-cranks (check the various photos on that page) work for various ages, but take a bit more trouble to fit because they affect the drive train. Until I found that link, I didn't know that you could get them for newer, fat tubes.
Crank shorteners are more suited to older children, and are really quick to fit. There are two related reasons to shorten a crank - both to put the pedals nearer the rider, and to reduce the circle described in pedalling, as a big circle could be a bit much for short legs!
As children get older, you can gain the odd inch or so of extra reach by using an old-fashioned seat pin and separate saddle clip; inverting the clip to sit above the saddle rails, instead of below, moves the saddle down (because the clip is always at the same height on the pin of course). So, if the adult tandem isn't too large, you may be surprised how early a child can lose the crank shorteners.
Obviously an adapted adult tandem is more likely to meet your requirement of riding with your partner. On the other hand, if you can afford a purpose-built child-back model, that's easier. Bear in mind that stokers (even adults) don't really need to be able to touch the ground in the same way as they would on a solo, as the steersman can easily balance the stationary machine.
I don't have experience of riding with very young children. We didn't do much till ours were over 7, and then it was limited. Therefore, we only ever used crank shorteners. I have seen set-ups with
side rails and all sorts.
Popular cycle-hire places, such as at reservoirs and railway paths, often include tandems in their fleets. You might do well to go along and try a day's hire as a learning exercise?