The bee population in your hive(s) should be increasing greatly at this time, even in new packages. This population increase should coincide with the honey flow which should be starting about mid month and continue through July.
Your bees will be at their busiest for the next several weeks as they take full advantage of the season and gather all of the honey, pollen and propolis they can. Encourage your bees to grow and increase as this will help you have a larger honey harvest.
Check the hive population, you want it to be steadily growing and strong. To check the population, observe the bees flying in and out of the hive; if you can easily count the number of bees the colony may be weak or compromised A strong hive will have between 30 and 90 bees arriving/leaving the landing board per minute.
Watch for signs of honey flows and ensure the bees have plenty of room to grow and store honey. If you do see signs of disease or parasites remember to NOT medicate until the honey has been pulled off.
Other tasks to consider: keep adding supers as needed to prevent swarming (remember to reverse honey supers) and don't add too many to soon or they may not fill completely; check for diseases; check for a strong queen (eggs and brood) - requeen if needed; make sure you have enough honey supers and frames to last through the summer.
There is much for you to do this month. One of the key items for this time of year is to keep the hives cool; there are many ways to do this:
a)shade the hive from noonday sun with fencing, boards, shrubs or shade breaks
b) stagger the supers slightly to increase air flow throught the hive
c) make sure fresh water is available at all times,
d) bore 3/4" holes into honey supers (upper corners) or
e) space the inner cover up off of the top super by a little (not too much or they make make burr comb).