Why and when are bonds broken
Is the breaking of bonds by using energy a prerequisite for new bonds to be made with the involved atoms?

A: No, it is possible for a bond to form between two species without first breaking a bond. But it is more common for reactions to occur with a tradeoff between breaking old bonds and forming new ones.

If so, is this because the more energy there is, the more the atoms will move/vibrate, eventually pulling the bond apart? 

A: Adding energy in the form of heat can lead to breaking of bonds, which in turn can lead to formation of new bonds. 

Otherwise, can bonds be broken by a new bond sort of pulling it apart in some way?

A: You can have a "displacement" reaction where one atom replaces another in a structure: the bond to the new atom forms at the same time as the other bond is broken. That process has a lower energy barrier between reactants and products, relative to breaking an existing bond first and then forming a new bond later.

Is the strength/length of the bond what determines the amount of energy needed to break it? I'm also unsure of what the strength or length of a bond is based on.

A: The strength of a bond is the same as the amount of energy needed to break it (in the absence of other reactants). Bond length tends to correlate inversely with bond strength, as for example in going from a single to a double to a triple bond between carbon atoms. Bond strength depends on a number of factors--bond strength tends to be greater between two atoms of about the same size. Bonds between two electronegative atoms like oxygen or chlorine are weaker than a bond between two carbon atoms. 

Am I correct in thinking that bond order is about the number of shared electrons in a covalent bond? But what about ionic bonds? Is it the amount of electrons "taken" by the more electronegative atom?

A: You are correct here in your understanding of bond order. But the term "bond order" does not apply to so-called ionic bonds -- for example in sodium chloride -- where an electron is completely donated from sodium to chlorine and does not participate in keeping the atoms together. Rather, it is the electrostatic attraction between the sodium and chloride ions that keep them together.