During this time your baby isn’t going to need much in the way of entertainment, but it is nice to have a place that you can put the baby when you need your hands-free, and it’s a bonus if the baby actually enjoys being there.
Personally I think a bouncer is pretty universally appreciated item of baby gear. This is a little baby seat that is a bit springy so as they get more active they can actually bounce themselves by kicking, or you can bounce them by using your foot. Even really basic ones now have vibration or some music built in which your baby may or may not appreciate (ours didn’t). When they’re really little they can sleep in the bouncer and also hang out and play there. A lot of bouncers now have all kinds of souped-up toys built in, either in a toy-bar that goes across the seat, or on a mini-mobile. At least a basic bar is kind of nice because then you have a place to clip your own toys on as the baby’s interests change. I wouldn’t go out of my way to buy the most expensive bouncer though because really they just need a comfortable place to hang out that’s mildly entertaining. We also had a variation on the bouncer which was called the “baby papasan” by fisher price. It didn’t bounce but was just a really soft and cushy little angled seat for babies. Henry loved this thing and spent a lot of time in it, especially at the beginning. He had reflux and had to sleep at a 45 degree angle which the papasan chair provided. We just put him in the chair and put it in his crib at night. If you have two floors, it’s nice to have at least one place on each floor to put the baby, so you might think about two bouncers, or a bouncer and a papasan-type chair.
Lots of people swear by a swing for babies and tell you they would have been lost without them. Most babies are lulled asleep by the rocking motion. I believe almost all (if not all) swings that are sold today are battery operated (maybe some with ac power) so you don’t have that annoying feature our parents had of having to re-wind the swing up manually just as the baby is about to fall asleep. I will say that not ALL babies love the swing and ours was one of them. He tolerated it kinda. The main thing I would suggest is buying a swing marketed as a “travel” swing. These are usually low to the ground and fold up. I can’t imagine actually traveling with it, but it takes up a helluva lot less room than a full-size swing, especially if you don’t know if your baby is really going to love it. They make fancier swings that are almost like a cradle—they go from side to side and back and forth. If you’ve got room for a full-size swing and fall in love with it—go for it, but if none of those apply just get the most basic one you can find. One thing to consider—it’s nice to have a swing that is either open on top, or that has a bar overhead that flips out of the way so that you can get the baby in and out of the swing without having to contort you or the baby so their head doesn’t get bumped.
Another crucial baby lounger/entertainer in my book is the baby gym. This is generally a square playmate with bars that criss-cross over the mat which hold a few toys that dangle down. The more basic models are really designed for the baby to lay on their back and have stuff to look-at/reach for. I’ve seen a lot of the newer/fancier ones that have more activities built-in that would apply if that baby was on their stomach as well. I think this is a feature worth considering because the babies need to be on their stomachs a good deal, and the more entertainment they have while they’re hanging out in that position the better. I have noticed they sell separate “tummy time” mats, but this way you wrap two up with one.
For the first 3 months, babies don’t need a whole lot in the way of “toys” because they’re enamored with each new thing they can actually focus on and see. They’ll love your face, looking at the light, watching things held close to them etc. It’s nice to have a few rattles and some babies really dig the little wrist and ankle rattles that babies can wear. They also make socks with rattles built in called “rattle toes” which are cute. A mirror made to stand up that baby can look into when they’re on their stomach is often a winner and some babies like really soft little stuffed animals this early as well. Most of the toys marketed at this age are soft and they make some kind of little noise- a crinkle or a squeak. There’s lot of cute stuff so I would register for some and just see what you get. You’ll always be able to fill your collection in once your baby is here and you learn what they really like.