"One thing that I've been wondering about is the logistics of holding both babies when they're newborns and can't hold their heads up yet"
You'll often be holding one or the other, but not both. If you do need to scoop them both up, you make sure they're side by side somewhere safe, like in a crib or on the floor, or way in the middle of the bed with you absolutely right there. Then you lean over, put your hands under their bottoms and sort of push your arms up toward their heads until their heads are pretty much in your hands. It really is a scooping motion. Then you lean down further until your body is against their bodies, and then straighten, all the while holding the babies against you.
Needless to say (but I'm saying it anyway), it's not the optimal way to move newborns around. If you need to move babies from their bedroom to the family room, for example, you do it one baby at a time, always making sure you have a safe place to put them. A floor is a safe place if there are no safety hazards, other siblings, stairs to fall down, pets, etc. Other safe places are cribs, play yards, swings, infant seats... When you're moving, one at a time is safest and using two hands is safest.
I also wonder about getting them both in and out of the car -- do I have to leave one baby in the car while I bring one into the house, then go back out and get the other one? Or will I get adept enough (and develop the upper body strength) to scoop them both up at the same time? (I'm talking, again, when they're too little to hold their heads up; after that, I think I'll be good.)
Good question. It all depends on your particular situation. If you have a garage, you drive in the garage, close the door, and can take babies one at a time. Same with a driveway if you're in a safe neighborhood and wherever you're taking the first baby is right inside the door so you're not out of sight of the car. This also depends on how paranoid you are; I'm the queen of paranoia, but there are others who are comfortable with leisurely taking one into the house, settling him/her in a back room, and coming back to the car two minutes later without any view of the car in that time. I'm the "better safe than sorry" type.
I expect you'll have the kind of car seat for your newborns that has a base and a seat that comes out of the base. Usually people carry babies from the car in the seat; this is what I recommend. Yes, a seat and a baby are heavier than just a baby, but this gives you a lot of flexibility. You can take both babies in their seats out of the car and put them on the lawn. Now you can either carry them both to the front door, or carry them in stages, but in any case you can make sure you're not too far from either of them. You can lock up the car and get your bag before you carry babies, go back to do it, whatever. Meanwhile, they're safe in their seats and not able to get into trouble.
It's not that you can't carry babies individually without their seats, but I don't recommend it. Trying to get two babies at the same time from a car to the house without their seats would definitely be unsafe, though. How do you unlock your front door if they're not in seats? The seats can generally be carried in your hand or in the crook of your arm.
Aso depending on your situation, you could transfer the baby seats to a stroller system, roll the stroller to your door, and then move them from that into the house one at a time, of transfer the babies themselves to a stroller and do the same.
Another reason to bring the babies into the house in their car seats is you'll want to be able to do the opposite when you need to take them somewhere. You get them ready in the house, put them into their carseats, then move them to the car. You don't want to have to run out to get your car seats.
Those bouncer seats, whatever the brand or style, are invaluable for having a safe place to put a baby when necessary. Later, when they can hold up their heads, something like an Exersaucer is wonderful. We had two downstairs and one or two upstairs, all bought used at a Mother of Twins sale. Swings are good for younger babies, but some kids don't like them, so start with one. The jumpers you put in a doorway are for babies who hold up their heads, similar to the exersaucer; they provide some variety for the babies. Again, only one of those are needed.
I would never (ever, ever) recommend a walker. Even if you don't have steps in your house, just don't do it. It allows too much mobility. Say you have a visitor who set a bag down? That bag is now within reach of a baby. Also, maybe you're careful never to take the walker anywhere unsafe, but your husband, mom, sister, whoever else is keeping an eye on the baby wouldn't think about it. I know a child who took a tumble because Daddy brought the walker onto the back porch...yeah. Luckily, the baby was fine.
I had a great play yard for my kids, the Today's Kids play yard. It looks like they don't have it anymore (except used on Ebay), but the Li'l Playzone with extensions is similar. We used this all the time once the twins got more mobile; it's also a great place for newborns if you have older kids or pets. It was my singleton's safe place as a baby -- a place where the four-year-old twins wouldn't trip over him.
Getting out and about
In the grocery store, I pushed the double stroller and pulled a cart behind me. To do this, I really needed a stroller with a horizontal bar. With my left hand, I grasped the bar toward the right side, and then placed my left forearm along the bar. I was able to push and maneuver with one arm in this way, and then to pull the cart.
There are double carts and carts with benches and fire truck carts. None of those work for little teeny babies; they need to be sitting up, holding their heads up, etc., for those kinds of things. Also, you park in the parking lot and then…what?...leave the kids in the car and run to get the cart? I do know a triplet mom who used to do this, but I don’t think it was optimal – she had little choice, probably.
For things like mall trips, usually everything I bought would go right under the stroller. Big baskets under your stroller are very nice to have.
Later on, as my kids walked, they had harnesses. They really will go in two different directions, and as quickly as possible; the harnesses were invaluable to me in keeping my kids safe.
There are obviously a lot of things to think about when you have twins. I remember mothers of singletons saying they'd "just" do this, or "just" do that. For example, "Oh, if he's having trouble settling down for his nap, I just pop him in the stroller!" I would sometimes say "But what about the other one?" That mother didn't have to worry about messing up the other one who was having perfectly lovely naps. Parents of twins don't "just" do anything.
You definitely will find the things that work for you and work for your babies. You'll find your stride.