Here are some things not covered in other answers:
In larger cities, the Greyhound stop will be an actual station, where you will go inside a terminal, find the correct gate, and get on a bus. Unlike airlines, you do not need to "check in". Generally the departure gates are fixed by destination, so a bus going to a given destination almost always leaves from the same gate. There are exceptions to this, though, such as for mechanical issues, crowding, etc. And in some cities, such as Chicago, a bus might depart from any gate. Here you will need to check gate assignments. As noted before, you can look up station addresses on the Greyhound web site, though they aren't hyperlinked from your itinerary.
You can purchase your tickets all the way up until seconds before departure, (and I once purchased tickets a few minutes after scheduled departure as the bus was late) though the web site will not sell tickets after 2 hours before departure. Also, you can get discounts for purchasing in advance. Typically two weeks or a month in advance gets you the cheapest fares, but discounts can be available all the way up to the day of departure.
On that note, it's not strictly necessary to be very early for the bus, unless you didn't pay for priority boarding and want a choice of seats. Greyhound recommends an hour before departure so you will have time to spend money in the station shops, and a bare minimum of 15 minutes before departure to board the bus, and that seems fine to me.
You can bring luggage onto the bus. There are overhead racks which will hold airline-sized carry-on bags. You can also bring a personal item such as a laptop bag. En route, you might also purchase food at various stops and bring that aboard (see below).
When you are transferring buses, you do not need to get your luggage out from under the bus. The driver will do that. However, you do need to take it with you from the side of the bus to the side of the next bus.
En route, the driver will announce all stops. If the stop will be more than a few moments for pick up and drop off, he will also announce how long the bus will be stopped there. At major cities, the driver will also usually announce connecting routes, though that's not always practical.
One thing that's not obvious to a first time traveler is that long distance buses will occasionally make rest stops from 15 to 30 minutes. There is one scheduled on your particular itinerary, at Benton Harbor, Michigan. These are an opportunity to spend a bit of extra time off the bus, and usually also to get some food. This particular location is at a Flying J truck stop, so you should expect to be able to grab some takeaway breakfast here. You likely won't have time to do so in Chicago or Milwaukee, so take advantage of it.
Another thing is that Greyhound tickets for other regional bus companies, such as (in your case) Jefferson Lines. You'll be on a bus from a different company on the last leg of your trip. Therefore, the services offered on board may vary. For instance, there may or may not be Wi-Fi, power outlets, or a movie screening. But the basics will work the same: There will be a restroom in back, and the bus will go down the highway.