A great way to start is with the KHS Tandemania Sport. We have had one for three years, and although it is heavy, it is great. The cost is under $1000.00. We have had a lot of fun with this bike and are getting ready to get a new road tandem. We did a couple of modifications like the seats and stems. I can ride it solo, when I pick my husband up from work, (I say a lot of prayers that I will find a curb to climb back on the bike at stoplights, as he is 5'11" and I am 5'6"). the ride home is a sure delight. I always have some stranger put their thumb out at me on the trip downtown.
Try and ride one of these, they are great especially if you want stability.
Diane