Identifying functional groups
Some pointers:


  the hexagon structure signifies an aromatic


Not quite. The hexagon structure depicted with alternating carbon-carbon double and single bonds is aromatic. It is called benzene and is very commonly encountered in organic chemistry. It can be represented by two resonance structures as shown below and is sometimes also represented with a circle inside a ring of singly bonded carbons to show the delocalised $\pi$ system.



However, the are other molecules depicted as hexagons which are not aromatic such as cyclohexane.



Additionally, there are other molecules which are aromatic which are not benzene, and many which are not hexagonal but you probably won't come across these at an introductory level.


  My first guess was carboxylic acid, but with further googling, I'm now thinking it's an amide, because it contains nitrogen and oxygen


You are correct that it is an amide. The amide functional group contains a nitrogen adjacent to a carbon-oxygen double bond. The carboxylic acid group is similar but the nitrogen is replaced by an $\ce{-OH}$ group. Amides can be primary, secondary or tertiary depending on the number of carbons that are bonded to the nitrogen.




  alkane is $\ce{CH4}$, alkene is $\ce{H2C=CH2}$, and an alkyne is $\ce{HC#CH}$


These are the simplest example of alkanes, alkenes and alkynes. They are called methane, ethene and ethyne respectively. However, there are more complicated molecules involving these functional groups. In general, alkanes have only carbons-carbon single bonds and carbon-hydrogen single bonds. Alkenes are distinguished by having a carbon-carbon double bond denoted by two lines between the two carbons which look like an equals sign. Alkynes have a carbon-carbon triple bond which is denoted by three lines between the carbons. Therefore, your example is an alkene.