What might be causing my battery to drain so quickly, and is it time to replace it?

Today I noticed my phone battery draining very quickly, perhaps even by 4% in the span of 2 minutes. It also seems to be taking a bit longer to charge. I'm not sure what could be causing this, but was wondering if this sounds typical of when a battery needs to be replaced.

Is it worth getting a battery replacement over a new iPhone? Will replacing the battery harm the data on my phone (i.e., should I do a backup before the repair?)