Why should we study all the three forms of representation of finite automata?
Add regular grammars for a fourth. There are others...

Part of the interest in DFA + NFA is that they are simple computation models, with NFA (and $\epsilon$-NFA) examples of nondeterminism (a crucial idea for more elaborate models). To prove DFA and NFA accept the same set of languages is also exploring a very important phenomenon in a simple, understandable setting.

Regular expressions (and also regular grammars) are completely different formalisms, that happen to describe the same set of languages. Again, the proof of this fact explores important cross-relations, and are an example that formalisms might look very diffent, be based on radically dissimilar concepts, but describe the same languages. Again, in a rather simple setting.

For "real world" use, you can start with a regular expression and get a minimal DFA for high-performance searching. Digital circuits are essentially DFAs, understanding them is central in computer engineering. Last but not least, often systems can be modelled as "being in a state" and "moving to another one" on external stimuli, even if the system is very far from a real DFA viewing it this way might help understanding it.

Added later: As noted by Raphael, it may be more efficient to interpret an NFA directly for searching, because creating a DFA can be expensive, and an NFA may be much smaller.