Is Splunk Light a way to learn on home routers with NAT. For instance, can you learn the Splunk basics with just the home network as your domain of users?
Yes definitely! There will be a few drawbacks though such as setting up alerts and not having the ability to do distributed searches, but I doubt you will need this in your case since it's a home network. I would recommend you learn how to use syslog data and stream it to a server then set up a Splunk forwarder to push it to your indexer.
You could go one step further and get a free AWS account which provides up to 750 instance hours per month and spin up a dozen VM's, set up 2-3 network tiers, configure the firewalls, and Splunk everything. This will also open the door to learning how to configure your UF's, indexer, deployment server, .CONF files, and search heads across different VLAN's
http://www.splunk.com/en_us/products/splunk-enterprise/free-vs-enterprise.html