What is the recommended method to backup 2 node Exchange 2013 DAG running on Windows 2012 Std? Currently we are taking daily full backup and keeping it for 2 days only due to space limitations. Can we take weeky full and daily differential or daily full is better like we are already doing?
This is going to depend on your DR requirements for exchange.
If you have the space available, yes daily fulls is the best route and fastest recovery.
If you have limited space then doing Fulls / Inc or Full / Diff works just as well. They will just take a little bit longer to do a full recovery with. GRT restores should take around the same amount of time (from disk).