Even in mid-May, park was covered in snow and lakes were frozen. So, don't expect your cell phone to work there. We also found limited vegetarian food selection but survived on pre-packaged food and grocery from nearby towns. For other three days, we had booked private room in The Hostel, in Teton Village ($64+tax=$73 per night) which turned out to be small basic room too far from the Grand Teton park itself! We essentially checked out after two nights and found one room in Billings on last day as we had return flights from Bozeman.
That's it for planning. Once inside the park, route is fairly straightforward with lots of maps to help you find path. You won't need GPS as there are few main roads and few junctions and you can just trace your journey all the way. Each spot is well marked and parking is freely available, though bit far in some cases. Most places require walking and aren't very accessible with wheelchair or stroller, and you will have to lift those on parts of route if you do decide to use.
Bison are plenty everywhere. Law requires that you don't honk at them or disturb them in anyway and maintain 25 yards distance, so if they do end in middle of road, you are supposed to wait silently for them to move away. That took about half an hour one time this happened with us! In practice, impatient driver do tend to try to pass beside them if part of road is accessible but you may be in trouble Park Ranger catches you.
We had unexpected park closure on third day of our trip and had to make do with trip round Teton Park since Yellowstone was not accessible. It turned out to be not bad decision at all and I will encourage you to consider drive around Teton for one day's plan (map). There is no extra fee for Teton Park if you've taken week long permit for Yellowstone. You can chose to decide which route to cover on which day, but in my experience three days are good enough for Yellowstone Park and one day for Teton Park.
As you can see from pictures below, Park is famous for its geysers which are spring of hot water and gases emanating naturally from ground. This gives unusual colour to ground and surface water due to bacterial chemistry. Popular Old Faithful geyser has predictable fountain spray every hour or so.
This has been one the most memorable trips for us!