There are various ways you can play this game, the "Go Slow" strategy is only one of them. Personally, I favor it because it allows for rapid expansion later on. Having said that, it is up to you to come up with a strategy that works for you.
Dwellers that are out exploring do not consume resources. Having said that, it appears that total population (including those exploring) governs when molerats and deathclaws start appearing. Rooms and levels of upgrading also seem to play a part. The way I play, molerats start appearing when my total population reaches 33 and deathclaws appear somewhere in the mid-60's (I think 64, but don't quote me on that). Killing and removing Dwellers to drop below the threshold does not seem to work, That is to say once molerats start happening, they will continue even if your total population drops below 33.
Here is why people say to work on resources first. Power is the most important. If your power drops into the red, rooms in your vault will start shutting down. If a room shuts down, it will no longer produce anything. Storage capacities of rooms seem to be unaffected when it is shut down. That is to say, if a Living Quarters gets shut down, that will not reduce the total population capacity of the vault, but you will not be able to use that room to make babies or hold dances. What you don't want, is food or water production to be shut down if power goes into the red.
If food storage goes into the red, all of your Dwellers who are not exploring will start to lose health and happiness. It should be noted that your Dwellers cannot starve to death, but a vault filled with dwellers that only have one point of health is just a disaster waiting to happen.
If water goes into the red, all your of Dwellers who are not exploring will start taking radiation damage and lose happiness. It should be noted that your Dwellers cannot die of thirst or die from radiation poisoning, but radiation poisoning will reduce the maximum health of a Dweller. That is to say, if a Dweller has a maximum of 100 health and gets 20 points of radiation damage, their maximum health will be reduced by 20 so they will only have 80 health and will not be able to be healed beyond that until the radiation is removed using RadAway.
Basically, if your food and/or water resources go red, then all of your Dwellers start losing health and they become less able to deal with any incidents that occur. The longer this state lasts, the less capable your Dwellers become. This is the basis of the Go Slow and Take Care of Resources strategy. If it doesn't work for you, then find another strategy that will.