How do I figure out what the model number is of my Shimano rear derailleur? Here are pictures of my current derailleur I need to replace.
Definitely one of the models in the Shimano Tourney range.
You don't need to know exact model to replace it. You just need to know what specs the replacement derailleur needs. You can look at the derailleur models in the Tourney lines in the Shimano Line-Up pages (you might find you exact model by comparing pictures also).
All the specs are pretty self-explanatory apart from speeds which is the number of sprockets in the rear cluster and total capacity needed which is tooth count difference between largest and smallest sprocket plus tooth count difference between largest and smallest chainrings (if more than one).