Replacing a dishwasher is a manageable DIY job that comes down to three connections: water supply, drain, and power. Getting the drain routing right is what prevents leaks and dirty water backing up.
Shut off and disconnect
Turn off the dishwasher's power at the breaker and the water at the supply valve (usually under the sink). Disconnect the old unit's water line, drain hose, and wiring, and slide it out. Have towels ready for residual water.
Make the connections
Connect the water supply line to the inlet valve with proper fittings, route the drain hose to the sink drain or disposal, and connect the power (hardwired or via a cord kit) following the manual. Don't overtighten plastic fittings.
Prevent backflow
The drain hose must rise into a high loop under the counter or connect to an air gap on the sink, so dirty sink water can't siphon back into the dishwasher. This is a code requirement in many areas and prevents smelly backflow. Run a test cycle and check every connection for leaks before pushing it in.