As always: All beekeeping is local. The following is based on my experience in the Midlands of South Carolina.
Plan on checks every two weeks.
Swarm season continues. Watch for swarms. (2013 – Swarm May 11th)
Continue to check for queen cells – make splits if swarm cells observed.
Monitor for disease and mites.
Remove honey supers that are filled. Provide super space for bees.
Notice Blackberries in bloom. Tulip poplar in bloom.
Then honeysuckle, Dandylion, Privit Hedge, Confederate Jasmine, Persimmon
May 1st – May 31st
1) check on need for adding empty supers at first of month but add additional
space conservatively toward end of month to make sure honey gets capped properly
2) Watch for swarms and capture, if possible.