How to Move the robot in joint and Cartesian Space?

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I would suggest going through these MoveIt! tutorials!

Edit 1(updated context): In the case of a UR5 robot, you would first need to look into the UR robot driver as it is essential for using UR robots with ROS.

For using Python, you would need to use move_group = moveit_commander.MoveGroupCommander() and with that you would be able to plan in the joint space like so (copied from the tutorial):

joint_goal = move_group.get_current_joint_values()

joint_goal[0] = 0 #[0] represents the joint of the robot, value is in radians

move_group.go(joint_goal, wait=True)

and in cartesian space like so (copied from the tutorial):

waypoints = []

wpose = move_group.get_current_pose().pose

wpose.position.z -= scale * 0.1 # First move up (z)

wpose.position.y += scale * 0.2 # and sideways (y)

waypoints.append(copy.deepcopy(wpose))

(plan, fraction) = move_group.compute_cartesian_path(waypoints, 0.01, 0.0) # waypoints to follow # eef_step # jump_threshold

As for using gazebo for that, this is also mentioned here!