Will I need a new case for these parts I ordered?

Well it looks similar the the case my girlfriend's HP desktop. If it is then it should all fit, however, you'll probably need to improve the cooling a little case fans)

Well, you could keep the case, or you could buy a new one, of which would have far superior cooling for your new components.
For This case: http://www.newegg.com/Product/Product.aspx?Item=N82E168...
And these Fans: http://www.newegg.com/Product/Product.aspx?Item=N82E168...
Total is $67.97. The Case Would have these fans in total:
Intake: 1x Front 120mm Fan, 1x Side 120mm Fan
Exhaust: 1x Top 120mm Fan, 1x Rear 120mm Fan.
Also, this case will be able to fit most future graphics cards, as it has enough space for 12.5inch graphics cards, such as the ATi Radeon HD5970. I would overall recommend that you do get a new case. I did. It helped!

I have little experience with upgrading computers, so would changing cases for my computer be something that is reletively easy to do?

^ I agree. If possible get a better case for sure.

Swapping parts from case to case is pretty simple and lots of fun, to be honest. As long as you can remember to put things back together the way they came apart its cake.

Your case looks like a standard atx case to me. If so, you should have no problem.
The psu you bought is a standard size, and the 6850 is not unusually long at 9.25".

If you like the case keep it.

As you remove the old psu, make notes of what leads go where. Or, take a picture before you remove anything to jog your recollection.

When you reinstall the new psu, do not force anything. Psu leads are keyed so that they only fit into the proper receptacles.

If your old video was using an integrated adapter, you should disable it in the bios, and point to the new card to display bios boot info. You will get a low res display on first boot. You will need to install the latest amd drivers to get your proper display resolution.

This is not rocket science, don't worry.