If you get your hands dirty, you become involved in something where the realities might compromise your principles.
It can also mean that a person is not just stuck in an ivory tower dictating strategy, but is prepared to put in the effort and hard work to make the details actually happen.
If you get your teeth into something, you become involved in or do something that is intellectually challenging or satisfying. ('Dig you teeth into' and 'sink your teeth into' are also used.)
Hand in hand= work together closely When people in a group, say in an office or in a project, work together with mutual understanding to achieve the target, we say they work hand in hand. There is no lack of co-operation and each synchoranises the activity with that of the other.