Some quick tips to keep in mind...
First off, know that your photos are not going to look the same as professional photos, and that's ok! But you are capturing some precious memories, and that is all that matters. Babies are energy-based and will pick up on your anxiety and frustration, so just relax and go with the flow!
Make sure the room is warm, then give your baby a good feed, and be sure to burp them, so they are happy and sleepy! But remember that awake baby images are beautiful too!
Your baby will probably poop or pee if you take photos without a nappy on, so have supplies ready for quick clean ups.
Don't worry if you don't have a DSLR - try putting your phone on portrait mode if you have it to get some yummy blur. Make sure your lens is clean!
And since you will keep things simple - you don't need any props, but blankets, teddy's etc. will make the session your own.
What to wear
Have the family wear comfortable, simple, neutral-coloured clothing. Stay away from logos, words, and characters on shirts, and neon colours. Newborns photograph best in a simple white onesie, as most newborn clothing are cute, but they don't fit well! You can add a little hat or headband for variety (but not one so big it takes the focus away from baby!)
Baby is more likely to stay sleepy and content when swaddled tightly. Try a simple swaddle wrap - learn how here!
Try to take your photos in the rooms that get the best gentle window light (not direct sunlight). North/south facing windows are great! If there's enough window light, turn off your overhead lights.
When positioning baby close to the window, make sure the light is coming from the side or towards the top of the baby's head.
Always have another adult present to be close to baby while taking photos with siblings. Start with low expectations - they may not cooperate and that's ok. If they lose patience, try those photos another time.
The safety and comfort of your baby is more important than any photo. Please do not attempt any poses that you see newborn photographers do - they are trained in newborn safety and posing.
Pets and siblings should never be photographed with a newborn without another adult right there to keep an eye (and hand!) on baby. When you are taking photos above baby, always wear the camera strap around your neck.
Listen for baby's cues. Your little one may start to tell you they have had enough. And pace yourself - you just had a baby and will need rest! If you need to break it up over a few days, thats A-OK!
Take loads of images so you can choose the best of the best. On your phone, VSCO is a great app for editing your newborn images.