This can be achieved by multiple ways. The easier and safest way is to create a replica on the new node. Replica can be created by starting a new node on the new server by assigning the same cluster name. (if you have changed other network configurations then you might need to change that also). If you have initialized your index with no replica before then you can change the number of replica online using update settings api
Your cluster will be in yellow state until your datas are in sync.Normal operations won't get affected.
Once your cluster state is in green you can shut down the server you do not wish to have. At this stage your cluster stage will go to yellow again. You can use the update setting to change replica count to 0 / add other nodes to bring cluster state in green state.
This way is recommended only if both your servers are on the same network else data syncing will take lots of time.
Another way is to use snapshot. You can create a snapshot on your old server. Copy the snapshot files from the old server to new server in the same location. On the new server create the same snapshot on the same location. You will find the snapshot file you copied. You can restore it using that. Doing it using command line can be a bit cumbersome. You can use a plugin like kopf which will make taking snapshot and restore as easy as button click.