With the infrastructure you're going to have sitting around your house, and your lack of advanced networking knowledge, you're not going to be able to sniff packets on your desktop using a laptop.
While your desktop is connected to your home router, traffic between your desktop and router will not be replicated over other router ports or over wifi. Traffic from your desktop gets routed exactly to where it needs to go.
While your desktop is connected to a home switch, the same logic applies--
traffic between your desktop and switch will not be replicated over other ports either (unless you ARP flood it and your switch is susceptible to this attack).
What you're trying to do would have been possible 20 years ago if you connected a consumer hub between your desktop and router, since traffic on those got blasted to every port, but hubs don't support modern network speeds so they've fallen by the wayside. The modern replacement is a network tap but you can't just buy one of those at your local Best Buy ($$$)
Given your environment you'd be best off generating pcap files on the source box and shuttling those elsewhere for analysis. Use daemonlogger instead of Wireshark for this sort of task; Wireshark is way too bloated and doesn't handle file rotation.