I guess the title pretty much sums up the question. This would be my first time out of the country, and so I'm a little unexposed to things like this (and will not have anyone with me). Will a JR Pass work just fine for traveling between Sapporo and Tokyo? Are there any caveats? Do you have to get the green car version or go by ferry or anything else like that - or book reservations - or can you just take the standard JR Pass and run with it on demand for a regular train between the two?
The quick answer is yes, you can. Read on for the details
What to get
You have to get your JR pass from your home country before you leave for Japan, they're not for sale there ! Here's a list of dealers, search for your region and buy it there.
The green car is not necessary unless you want it, it's like a first class train.
How it works
Once you have it, your pass is your train ticket, you just show it to board. The JR pass allows you to travel on JR lines (except Mizuho and Nozomi trains), so make sure to avoid the small private networks and you should be fine.
Some trains require reservations so you will have to go to the station to reserve a seat (you can't do it online). Your JR pass means the reservation is free, but you still have to do it. There's no minimum time before the train leaves, I've done it 20-30min or a few days before, it doesn't take very long, but if it's a busy route all the seats may already be reserved.
Japan Guide has a great page about this.
How to get there
Now, with all that said, there are many trains that run between Sapporo and Tokyo, especially now with the introduction of the Shinkansen you have more options, and there's always an overnight train too. You can definitely do the whole journey on a train only.
Hyperdia is your best friend here, just make sure you check the right options when you're searching for trips !