What is public transit like in Hyde Park? Is it easy to get around?

Relatively easy. You can live in Hyde Park without a car (though it'll be a bit more inconvenient than the North Side). For transportation to downtown, take either Metra or the CTA Jeffery Express bus. Do NOT take the CTA Green Line especially at night.

Get the Chicago Card for CTA, you can ride both the Redline EL train and buses. CTA may have its own shortcomings, but in my opinion are light-years ahead of other public transit in major cities.