A quick bit of exposition on dice rolling in view of your players:
Dice add a bit of randomness to heighten tension by creating the possibility of failure. For your players they are the arbitrator of “fair” and the “what happens.” The dice coming up good or bad is part of what drives the game along, standing in for “the fates,” “the will of the gods,” etc. For the player when they roll the dice they are engaging in a critical game mechanic that can lead to stunning success or hideous failure, modified by their character creation and development choices over the course of their campaign.
For the DM dice rolls are a way to generate randomness when you don’t have a plan or a don’t have a preference for how things happen. The party tripping a standard pit trip, that will take about 1/4 of their HP’s, is a take it or leave it kind of thing. It’s not critical to the story, there are no real pro’s or con’s of it happening. Rolling that check in the open, or sticking by what the dice actually read behind your DM screen has no real effect on the game in the immediate sense (yes those HP may be valuable later, but that’s surmountable by sticking healing potions/artifacts/whatever in as appropriate). Random encounter tables, random NPC names, random loot, etc. dice rolls are awesome. They can even be fun for the PC’s to watch in the open(as they root for a particular piece of loot to get rolled).
Actions that should/should not occur for your campaign to flow satisfyingly for all of you, are too important to leave to the dice. Story elements or major flavor items, again too important to leave to the dice. Heroic victory or death, too important to regulate to the dice. Am I saying you should ignore killing to hit rolls or damage rolls, not at all. Make the players death worthy though. Having a PC killed by a random guard just due to a series of bad dice rolls in a throw away encounter isn’t fun for anyone. Having the PC killed by the main baddie in the climactic fight of the current campaign? Now we’re talking.
The PC’s are executing an incredibly well done stealth infiltration of the bandits hideout? Don’t allow YOUR dice to arbitrarily derail their plan unless it works for YOUR plan for the session. If the PC fails a stealth/sneak/whatever roll with THEIR dice, then sound the alarm. Don’t sound the alarm, just because a random check you rolled comes up low, and it doesn’t have a good effect on the story line. Respect the PC’s dice roll, as it’s an extension of the player and their only physical act in shaping the story (well unless you’re LARP’ing). Ignoring a players die roll, is basically either saying the PC/player’s interaction good or bad isn’t “correct” for the story, or unwanted/unneeded for the story. Never give your players that feeling. Commiserate or congratulate, but respect the roll and your player for making it.
In combat you can let the game flow along the dice rolls most of the time, as well designed encounters should be challenging but surmountable. If the encounter is getting too tough, don’t be afraid to call out natural 1’s (or misses) on YOUR to hit rolls to help let “luck” be on the players side. If the encounter needs to be difficult for the stories sake and everyone’s enjoyment (sure playing whack a goblin can be fun, but you don’t want an encounter to be a boring exercise in dice rolling), start calling out high damage numbers etc.
In general for combat I never say what I “rolled” except for damage. Nor do I ever tell the party what their to hit numbers are. For new players, they can focus more on the storytelling and coolness, than min/maxing their combat.
In short treat YOUR dice as a way to interject randomness into the campaign, and prop to move the action along in the direction it needs to. The players are bound to their dice, as allowing them to fudge their numbers freely removes the tension and drama from using them (a limited number of re-rolls etc, can be a great house rule though).