I have a custom setting and this has some records in it. How can we move these records? Would custom settings be also available in apex data loader?
You could certainly be able to extract and insert your custom settings via the data webservice APIs and also via the Apex data loader.
Another often seen practice is to have an Apex code snippet or maybe even Apex method which can be called to setup custom settings and perhaps some minimal test data. This Apex snippet or method can then be executed via the dev console to speed up preparation of fresh dev and config only sandboxes. This is often a faster and safer workflow than doing the export-->import of custom settings.
To make your task easy, you could use the "Custom Settings Exporter" chrome extension. It extracts all the custom settings from your org into an excel. This excel could then be loaded via dataloader (you might have to save each sheet as csv)