I realize there are a few guests that have not explored a National Park like Yellowstone before. For those who are coming on Monday I've dedicated this page as a reference for the trip.
This is an optional trip which everyone is more than welcome to join, but feel free to explore the park at your leisure. For those guests who would like to meet up. There are visitor centers at two entrances to the park I would recommend to start your trip at:
West Yellowstone Visitor Center (West Entrance) for those staying in downtown Bozeman and/or Big Sky.
Albright Visitor Center (North Entrance) near Mammoth Hot Springs for those staying near the venue or in Livingston.
I would highly recommend carpooling as there will be a lot of traffic going into the park this time of year.
That is completely up to you depending on your activity level. There are multiple places inside the park where you can get supplies/food. However, I would recommend bringing some food and snacks prior to the trip as there will be long lines and parking issues.
The park has very limited cellphone signal, radios are nice to have to talk in between cars. I would also recommend making sure you are at a full tank of gas before you go into the park as there are very few gas stations around the area, but other than that dress comfortably and stay hydrated. I've listed a few things below which you may want to pack prior to the trip:

For the guests who are more adventurous, yes you can obtain back country permits to do hikes further into the park by foot. Yellowstone has hundreds of hikes. The ranger station will issue these permits and you will be required to take a bear container for trash and spray and depending on the trail you pick. You will need to get the information on what you will need at the ranger station prior to any trail you plan on doing.
For the rest of our guests, the park is very well maintained and mostly paved roads. Hiking boots are not necessary unless you plan on doing a dedicated hike. It is however imperative that everyone, no matter what you are visiting, to STAY ON THE TRAIL. Do not go off any walking trail in the park for any reason! Yellowstone can be beautiful, but also very dangerous, there are geothermic areas everywhere with geysers which can spew water over 200 degrees.

Unless you're doing backcountry hikes, bear spray is not required. This should be an obvious one, but I cannot stress enough. DO NOT get close to any of the animals in the park or feed them! Regardless of what other tourists are doing, be respectful of the animals and give them space. Animals which get used to humans and approach them are put down as a liability.
Yellowstone is not a zoo, just because an animal eats plants does not mean they are friendly. These are wild animals and can be very unpredictable and violent especially during mating season. There are multiple injuries and deaths every year in the park from tourists not obeying the signs or things I've mentioned. Please do not be a statistic, get arrested, get fined, and enjoy the animals from a distance!
It is important to drive slowly in the park. Not only are bison and elk crossing the roads regularly, but coming through a blind turn at speed to have a 500lb bison standing in the middle of the road will be disastrous. There are huge clearings and landscapes which contain these animals if you look hard enough, I'd highly recommend bringing a good set of binoculars with you on the trip.