First of all, I'm a big fan of cycling. However, if being a pedestrian is working for you, why change? How would biking improve your lifestyle (other than making your husband happy)? Are you finding it takes too long to get places on foot? Are you wanting to go places that are currently too far away? Are you currently using transit, but want to stop having to spend money on a monthly pass? What problem are you actually trying to solve?
Second, getting doored is something you can entirely prevent by simply not riding in the door zone (i.e., stay far enough away from parallel-parked cars that their doors can't open far enough to hit you). Similarly, many other kinds of accidents can be either prevented or at least made much less likely by appropriate cycling behavior on your part.
In my opinion, the key is to ride predictably and
assertively: if you project the impression that you are a confident cyclist who knows your rights and the rules of the road, drivers will treat you respectfully. In urban areas, this predominantly means that unless you
want cars to pass you, you should ride far enough into the center of the lane to discourage them from doing so, aka "taking the lane." If you're worried about holding up traffic, get over it. The cars will wait or pass by changing lanes when it's safe to do so. Besides, urban streets typically don't have long distances where passing is difficult anyway, so you're unlikely to get anyone stuck behind you for more than a block or so.
In contrast, meekly trying to "stay out of the way" by hiding in the far right gutter, weaving in and out between parked cars, or worst, riding on the sidewalk, are good ways to invite a collision.
It will also need to tow a bike trailer if your kids are going to go with you.
A "long-tail" cargo bike or bakfiets (Dutch-style cargo bike) would probably be better options for carrying the kids, especially since the OP would be taking them "pretty much everywhere." IMO, a trailer is a better solution when its use is only occasional.
However, either of those two styles of bikes are expensive enough to be a serious commitment and uncommon enough to be difficult to find used. I certainly would not recommend the OP buy either one until she's sure she's going to stick with biking.
Cargo bike:
Bakfiets:
