As a vet the tips we give for people bringing a baby home with any pet are:
Get a youtube video of a baby crying and play on low volume every day, gradually increasing volume. Play in nursery and also in main room that pet lives in. This will acclimatise them to the sudden strange noises baby makes.
Have nursery pre setup and allow pet in room but don't allow it to sleep in room. You need the pet to realise that the room isn't that interesting and that it doesn't want to be in there.
Never leave nursery door open with a pet in the house! If you are going to use a baby gate for a dog start using it before baby arrives. Remember cats can sneak into a room and hide. My mum got a hell of a shock when day when she went into the nursery and found one of the cats in my bassinet with me, the door was closed!
For a cat get a Feliway diffuser, for a dog get a DAP diffuser. These release pheromones that make the animal feel safe and secure. You may need several round the house. Can buy online or pet shops or vet.
Give the pet a safe space somewhere else in the house that they can go to, it won't just be the noise of the baby but also all the visitors that can upset them! If you need to lock them in a room/crate then get them used to is before baby arrives.
Any changes to their routine should be instigated before baby arrives. If walks or feeding times will change then change them at least one month before.
The washing machine is going to be on a lot, make sure your pet is fine with the noise. If your cat doesn't like it make sure their litter tray isn't near the washing machine! they might be fine with using it if the machine is only on every few days, they won't if it is on every day and you will get accidents in the house.
Never ever leave a baby/small child with a cat or a dog. They can get jealous and they can be vindictive! Animals are far more intelligent then they are given credit for! It is their house too and they were there first!
Be very careful of animals knocking over bassinets. One of my friend's small dog would take a run and a jump at it as soon as it was allowed in the room! Let the pet first see the bassinet with baby smell without baby in it, so you can see how it will react.
Don't suddenly dangle the baby in front of them or invite them to come and sniff it. Take a cloth wipe it over the baby and then take the cloth to the pet in another room where they feel safe, allow them to get used to the baby smell that way. The first time they are in the room with baby and you, pay baby no attention and full attention to pet. The baby should just be there but isn't the main focus, so only do when napping and somewhere pet can't reach!
Make sure someone in the house puts aside some time for the pet, even if it just 15 mins a day that is just their time, where they get cuddles and attention.
Hope this helps and doesn't alarm anyone too much.