For educational purposes, I want to see what packages are sent over my home wifi with VPN on and off. What program should I use for this?
Wireshark or tcpdump would both be useful if you want to dump packets going across the network all you need is to put an interface into promiscuous mode and you should be able to see packets flying across your network. There are lots of tutorials on using wireshark and tcpdump in the interweb.
monitoring outgoing network traffic this question also appears to address what you are asking with regards to tools for examining network traffic.
In addition to previous answer be aware that not all Wifi Chipset allow you set the them in monitor mode (promiscuous mode). So first check what chipset you have: