What is a sign that my chain needs to be lubricated again? I just cleaned my gears, derailleurs, and chain. Lubricated the chain with Teflon dry lube. When will I need to re-lubricate again? Will I hear the chain make sounds? Do I do a visual inspection?
If you hear your chain squeaking you definitely need to re-lubricate it.  It's also a good idea to re-lubricate after riding in wet weather since the combination of road grit and water tends to strip away the oil/wax.

For more detailed recommendations, see this question:  "How often should I lube my chain?"