Distinguishing atom types
To identify hydrogen bond donors: Look for a polarized hydrogen (typically bonded to O, N, S) 
Hydrogen bonds form from the interaction of a partially positively-charged hydrogen and a (partially) negatively-charged atom. Thus, to "donate" a hydrogen for hydrogen bonding, you need a polarized hydrogen. Hydrogens are polarized by being bonded to electronegative elements: in organic molecules that will typically be O, N, or S. 

To identify hydrogen bond acceptors: Look for atoms with electron lone pairs, or otherwise partially negatively-charged sites on the molecule. (Typically O, N, S, or halides).
To form an interaction with a polarized hydrogen, you need a source of (partial) negative charge, which typically can come from electron lone pairs. The most common species you'll encounter in an organic molecule include O, N, S, or halides (F, Br, Cl, I).