There's honestly so much I could have done had essential oils been a part of my life when my son was a newborn, baby, toddler and "youngster." Super thankful we've had them since he was 10, but Newborn - I wish I had oils when...
First there's the doTERRA Baby Collection. I would have stocked up on this hair and body wash, diaper rash cream and baby lotion. As of July 2019 it's been out of stock for a while, but when it comes back in I'm snagging a few more because they also make the best baby shower gifts. 
A product I swear by, even without being a new mama, are the Lifelong Vitality (LLV) Supplements. These have been absolutely life changing for me, so I can't even imagine how much better I would have felt and what better sleep I would have gotten if I had these back then. The LLV Supplements are designed to supplement our Western diets and provide our bodies with the right dosing of vitamins, minerals and nutrients that we need - that unfortunately we're just not getting for our diets. This is due to the nature of the soil as well as the amount of food we can possibly consume, not to mention if you add processed foods into the mixture, because sometimes the last thing on our minds as a new mom is cooking. The reason I'm bringing them up in this post for newborns is because our babies depend on us. Especially if you chose to nurse. Because these supplements will help you with sleep and getting the intake of properly balanced goodness - it will help you and your body be on point for the day-to-day of being a new mama.
Diffuse. Diffuse. Diffuse. I'm going to recommend some oils to apply topically to your baby, but if you're like most who haven't had much exposure to essential oils, then start out diffusing them into the air. They will still benefit baby and the entire household really, but it's a safe (in my opinion fool proof) way to start.
I would have totally had the doTERRA Touch™ Kit. However the one I will recommend instead is a new one (so it wouldn't have been available to me back then) it's the doTERRA Kids Collection. It's absolutely perfect! These oils are already perfectly diluted and formulated specifically for developing minds, bodies, and emotions. These essential oil blends feature unique combinations therapeutically balanced to provide powerful benefits while being gentle on delicate skin. Check out the blog spotlight that will tell you the oils that come in this kit and how to use them.
Honestly I could go into a ton more, but I don't want to overwhelm you more than I probably already have. If you want some more suggestions, check out doTERRA's blog post on Using Oils in the Home: Nursery.
It would be an honor to help you and get you started as early as possible, so you're not like me looking back (and writing a blog post) saying I wish I had oils when... I'm here when you're ready!
Good luck mama, you've got this!
xoxo Danette