We received a lot of gifts prior to Sadie's birth. I registered for most of the things based on recommendations from friends or from researching online. 3 weeks in, these are my favorites so far (keep in mind we haven't even been able to use a lot of the stuff yet).
Fisherprice Snugabunny swing and Snugabunny bouncy seat
I'm sure just one would suffice, but I like having options. If she is not happy with one, the other will usually work to calm/entertain her. The swing can be set up to go in different directions, has a moving mobile and also makes different noises. The bouncy seat vibrates and also makes noises.
Graco Pack n Play
The one we have has a "newborn napper." We've been able to keep this in our bedroom and have a snug place for her to sleep at night. We're using the Prince Lionheart Diaper Depot to keep essentials organized in our room for her (diapers, cream, pacis, etc). It fits on the pack n play.
Summer Infant SwaddleMe
We use these each night to turn little girl into a burrito baby and to protect her from her flailing arms. These make it so easy because there are flaps with velcro. (Not a picture of my little girl, just the image from BRU)
Aden and Anais Swaddle blankets
These are great light weight blankets. We will swaddle her with them sometimes during the day or use it as a cover up blanket.
Cloud-B Sleep Sheep
We also got the Graco Sound machine, but just haven't had as much of an opportunity to use it. The sleep sheep has velcro on the back to attach to the crib, but so far we've mainly used it during the day. The sound that lulls her to sleep/keeps her asleep is the sound of the mother's heart.
Baby feeding log app
I was having trouble remembering when/how long she ate, how long she slept and when she was changed (possibly due to a sleep deprived brain...) I actually bought an itz been timer, but never could find it to log her info in when I needed it. I found this app and love it. I can locate my phone most of the time.
We have the BOB Revolution stroller for the neighborhood, walking outside and eventually running (when she is old enough for that). If you buy the BOB, go ahead and get the single handlebar console so you will have a place for your keys, etc.
I leave the snap and go stroller in my car and have used it to go in stores and doctors offices. It is quite a bit smaller so its easier to maneuver in smaller spaces.
Other essential items I've found are burp cloths and shout stain remover