A molecules polarity depends primarily on
1. The extent to which it can hydrogen bond
2. The number of electronegative atoms
3. The polarizability of the bonds or atoms
4. The net dipole moment of the molecule.
Applying these rules you can make a list for determining the relative polarities of the functional groups when all else is similar.
Alkanes are the least polar. No hydrogen bonding, electronegative atoms, polarizability and the net dipole moment is small at best.
Alkenes are similar to alkanes in all respects except polarizability. The p orbitals of are large and polarizable, making this class of molecules more polar than alkenes.
Conjugated Polyenes and Aromatic Compounds are similar to alkenes but now the pi system (of p orbitals) is even larger, increasing polarity.
Ethers and alkyl halides are more polar than the above classes because of the introduction of a polar atom. Rules higher on the list take precedence over rules lower on the list.
Aldehydes, Ketones, and Esters have a large dipole moment due to the carbonyl. Along with the electronegative oxygens, this makes them more polar than an ether or alkyl halide.
Amines and Alcohols have the ability to hydrogen bond, the first and most important rule on the list. Although it is a close call, everything else similar, an alcohol would be more polar than an amine because oxygen is more electronegative (Rule 2).
Carboxylic Acids are the most polar functional group because they can hydrogen bond extensively, they have a dipole moment and 2 electronegative atoms.
Notice how large the difference is between esters and carboxylic acids-hydrogen bonding counts for a lot.
Obviously the rest of the molecule plays in to any decision. If you have a 30 carbon chain attached to an any group it will be significantly less polar, but this can give you a framework to make an educated decision. The rules aren't perfect, and sometimes lab experience can surprise you. Sometimes you have two compounds that you really couldn't make a good guess about because they seem so similar. In that case, you run the column and figure it out by NMR.