Energy in bonds

This is how I understand the link between bonds and enthalpy.
Bonds contain energy. So to break bonds requires an energy input, and to make bonds, releases energy. ATP is energy currency and when it breaks down to form ADP, bonds are broken but why is energy released rather than needing to be inputted?
However, in a thermal decomposition reaction, a substance, such as calcium carbonate requires energy to break down, it decomposes only when heated, because energy is needed to break the bonds. This is why things don't spontaneously decompose to their simplest parts.
From another point of view, it's because atoms 'want' to have a full outer shell. Surely that would mean compounds with bonds contain less energy, because nature favours things with lower energy, but then bonds contain energy, so this doesn't quite fit. What have I missed here? I know we're going to do this in more detail in Year 13 when we do the Kreb's cycle, but is it the making or breaking of ATP that releases energy?
Is the enthalpy of something with bonds greater or smaller than the enthalpy of something without bonds?

New Resources

I am currently in the process of uploading new resources and aim to complete this by the end of this week. Please click on the links below to find resources for the new module that we have started this term, for both Mr Beaumont's/Mrs Braithwaite's and Mrs Sudbery's/Dr Everett's work.

http://cedarschemistry.edublogs.org/mrs-sudbery-and-dr-everetts-work/
http://cedarschemistry.edublogs.org/wp-admin/page.php?action=edit&post=93

Please let Mrs Sudbery know if there are any areas of AS Chemistry in particular that you would like some revision or practice resources for.