Unit tests are the responsibility of dev-team, not QA one. They normally run on service build phase and require a lot of special knowledge (real objects mocking for example)
What QA usually does is use the public interface to ensure the service provides the functionality it declares.
The most known tool for testing REST is SOAP UI, however since REST is acting over HTTP you can use any tool that supports HTTP.
You should also consider performance testing of REST. SOAP UI supports some level of performance testing, however I would still suggest to use Jmeter since it allows to build load scenarios in more flexible way.
What is not convenient is that REST services do not usually expose the interface model they use (unlike Web-Services which act using SOAP) so that you should either push your devs to provide you full specification of service interface or suggest them to use REST service building framewrok like SWAGGER or some alternatives of SWAGGER.
This will simplify the process and let you build test clients automatically on any change in service interface.