Just want to be sure that plugging the charging cable into one of the other ports isn't going to fry the machine
No you are not going to fry anything
You do not specify which exact mac model year you are using.
The different port sides only really come in to play with high bandwidth peripherals.
Late-2016 MacBook Pro models vary slightly in the data speeds they provide to each Thunderbolt 3 port.
MacBook Pro (15-inch, Late 2016) delivers full Thunderbolt 3 performance on all four ports.
MacBook Pro (13-inch, Late 2016, Four Thunderbolt 3 Ports) supports Thunderbolt 3 at full performance using the two left-hand ports. The two right-hand ports deliver Thunderbolt 3 functionality, but have reduced PCI Express bandwidth.
MacBook Pro (13-inch, Late 2016, Two Thunderbolt 3 Ports) delivers full Thunderbolt 3 performance on both ports.
https://support.apple.com/en-us/HT207443
https://support.apple.com/kb/SP748?locale=en_US