Eagle has recently made a few very notable improvements in the UI department but there are still some counter-intuitive things about how it expects you to work.
Origins are one of the most important things to understand in Eagle because just about every operation you'll want to do on a component will require you to interact with the Origin. The rest of the symbol or footprint might as well not exist as far as Eagle is concerned, you must click on the origin.
Some times they're easy to find:
Other times you'll click around like a maniac trying to find it:
And other times you'll swear you've gone crazy and start questioning if you've ever known how to use Eagle, only to later notice that you've hidden the tOrigins or bOrigins layer that controls if the origins are visible and can be clicked on:
Chances are that if you're trying to do something reasonable and Eagle is beeping at you or refusing to listen, you're probably not interacting with the origin.
As I mentioned previously, since being acquired by Autodesk, the Eagle team has recently made some notable improvements in the UI department. Previously some very common user interface operations like moving or deleting groups, copying and pasting were annoyingly difficult and unintuitive. This improvement in usability is important enough that you should verify that it is enabled:
With a board or schematic open in their respective windows, click the Options menu > Set..> Misc tab and double check that the "Group command default on" option is checked.
With this option enabled, two notable group operations are much easier. To move a group, all you have to do is select the Group tool, left click drag a rectangle around the items to group, and then simply drag on of the group's members' origin to move the group.
Similarly, after using the group tool to create a group, simply pressing the Backspace (Delete on a Mac) key will delete it.
Additionally, CTRL/CMD-C and CTRL/CMD-V can now be used to copy and paste selected items or groups. Note that, if you copy any nets, the copies will still have the same name and thus will be connected together.