Their goal is to reduce inflation. Increasing unemployment, is a side effect of rate hikes but ultimately, reduces demand for consumer goods thus reducing inflation.
Their goal is to keep inflation steady at a manageable rate, and to prevent any deflation at all. Since inflation indicators haven't been too reliable in the last 20 years, they were caught horribly off guard. But ideally, they would make small changes one way or the other to maintain a nice equilibrium that would keep employment levels relatively stable.
There are a ton of levers (not all under the Fed's purview). This is just how capitalism works though, others must suffer at the expense of capital. It's bad enough when the government doing the bare minimum, like trying to put a bunch of people to work to replace our crumbling infrastructure, gets decried as "socialism". Those voters must not understand (or simply not care) that voting for the same corporate interests over and over again will just lead to the working man sacrificing more to big business.