In the Purves book, tyrosine is listed as having a hydrophobic side chain. However, in the Becker text it is listed as having a hydrophilic side chain. I thought it would be hydrophilic due to the -OH group, but does the ring change that?

The tyrosine side chain has lots of CH groups, making the ring and stem very hydrophobic, but it also has that hydroxyl, which is actually VERY polar. So take your choice; Profs Purves and Becker picked the part they thought was most important and disagreed. My opinion is that tyr can be both: when approached from the stem side by a non-polar group, it could make a hydrophobic interaction. But its hydroxyl could also H-bond very well to a partially negatively charged group it points at. So I say, both. Nature need not be simple (or unambiguous).

How can I tell which side chains do what? Normally I'd just memorize without asking but it's sort of a tricky task. Some R-side chains with amine groups ionize to become basic whereas others do not. For example lysine vs. asparagine.

The asn (asparagine) side chain is not an amine, it is an amide: -CONH2 is not equal to -CH2NH2. The connection to a carbon bearing a double-bonded oxygen changes the basicity of the NH2 so that it does not ionize. You do have to recognize all the functional groups listed on the handouts, and recognize that an amide is an amide and an amine is an amine, etc. and which are ionized, which are polar, which are hydrophobic, the significance of the sulfhydryl, etc.