On your new hive Continue to check the hive for progress and queen activity on a 7 to 10 day cycle. If there are going to be any problems you want to see it in time to do something about it. That means your time between inspections needs to be less than the time it takes the bees to initiate queen rearing. You need to be in the hive in less than 10 days from the last time you checked.
Once you are confident the brood nest is healthy and stable, then move your inspections to be outside only. Especially once the population booms and the supers are getting stacked on. Observe the debris out front the hive and the bee behaviour and activity level around the entrance. Learn to recognize the difference between normal entrance activity and problems. Leave them alone if normal. Go in deep if you sense a problem. I continue to spot check when only one super is on. Once the second super goes on, I do not open the nest again until end of season when peeling off all the supers to reduce for fall/winter. Exception is if I detect a problem indicates by the bees and debris around the entrance.
Getting into the brood nest is easy with one super on. Worst thing to do though is setting it down once you have started the lift. Once you start to crack and lift, have a good grip and do not set back down. The risk is crushed bees and a crushed queen in bridge comb between the boxes.
1. two heavy puffs of smoke directly in the entrance
2. lift the lid. two heavy puffs of smoke under the lid. set the lid back down
3. wait 30-40 seconds for the smoke to spread and bees to settle down coughing on it.
4. crack off the super. Crack along the backside and tip it forward then lift straight up. Do not allow to set back down onto the bees, crushing them
5. set the super on the ground behind the hive, tipped up on end, frames vertical.
6. inspect the brood box as usual.
When done, smoke the bees down off the top bars of the brood box. smoke the bees up off the bottom bars of the super. put the super back on.
There is a good way for moving old comb out of the brood chambers. Similar but different from how max2 describes. If I understand correctly, your hive at this point is brand new and they are just getting doing drawing out on all new frames. You will not have to be concerned about old comb for 4 seasons.
PS: when you have the super off, also tip the bottom brood box up on end and look along the bottom bars of the frames. If they are advanced quicker than you think, you may spot the beginnings of swarm cells along the bottom bars. Let us know if that is the case as you may be needing some points of what to do pdq.