Are there any methods to help with memorizing the formulas of organic functional groups (ex. alcohol, aldehyde, ketone, carboxylic acid, ester, amine, & amide)?

A useful concept in organic chemistry is the degree of unsaturation. The following discussion may be longer than I like.

Alkanes have general formula, #C_nH_(2n+2)#, where #n# is the number of CARBON atoms. You should check this out. Is it valid for methane, for ethane (#C_2H_6#), for propane (#C_3H_8#)?

So in a FULLY saturated molecule, the hydrogen count is at a maximum. The energy content is also at a maximum in that #C-H# bonds can transfer a lot of energy. Each degree of UNSATURATION, each #C=C# or #C=O# or #C=N# bond, each RING JUNCTION, reduces the hydrogen count by 2. Where nitrogen exists in the molecule, we subtract #NH# from the molecular formula before we determine the degree of unsaturation: e.g. methylamine, #H_3C-NH_2#, is FULLY saturated by this protocol. How many degrees of unsaturation is possessed by acetylene, #HC-=CH#?

So, given a molecular formula, #C_nH_mO_n#, the first thing you check out is its degree of unsaturation. The tells you the number of double bonds and ring junctions for which you should be looking.

The way I do it is I relate functional groups by the methods through which you achieve each one. I ended up memorizing them as I learned them.

Specifically, here's how I group things together (there is some repetition to it, but I find it easier than #100%# straight up memorization):

1. #1^@# alcohol, aldehyde, carboxylic acid
2. #2^@# alcohol, ketone
3. carboxylic acid, ester
4. nitrile, amide, carboxylic acid
5. amine, imine, enamine
6. carboxylic acid, acyl chloride, acid anhydride

And alkanes, alkenes, and alkynes are simple enough that I do just memorize those.

I group the above functional groups like that because of the way they are related when synthesizing them. Here is roughly how I see it:

I remember the relationships between these functional groups by practicing the reaction mechanisms for each reaction that relates them, so that you get it down to muscle memory. What I would do is start practicing your mechanisms, and hopefully you'll start making some connections.

When writing syntheses, usually the hard part is making the connection between starting reactant and end product. You should be able to think such that you have more trouble with knowing how to get somewhere than with where you want to go each step of the way.