Does learning too much useless information weaken my memory?
Current evidence suggests that internet access is not weakening memory, but changing what information is prioritized.  This study referenced below suggests that when people expect to have future access to information, they are less likely to remember the information itself and more likely to remember where or how that information can be found.

Google Effects on Memory: Cognitive Consequences of Having Information at Our Fingertips
Betsy Sparrow, Jenny Liu, Daniel M. Wegner Science 5 August 2011: Vol. 333 no. 6043 pp. 776-778. 

DOI: 10.1126/science.1207745