5. Nightly Wasteland runs
You might play for a while before even realizing that the "Wasteland" is a place you can send your Dwellers. In fact, it's one of the more important places to send them in the game. The Wasteland is the source of weapons and gear for your Dwellers (other than the odd Lunchbox) - so sending Dwellers out to go scavenge is essential. The trick is, it take a long time. What I have found to be most effective is to gear up some of your best Dwellers (often the few uniques you get from early Lunchboxes) and send them out loaded with Stimpaks and Radaways right before bed. They may eventually die during the night, but they'll rack up so much gear they'll be worth reviving. A few nights of doing this, and your Vault will be wearing Wasteland designer fashions in no time.
6. Rushing rooms
My first glance at the "rushing" mechanic in Fallout Shelter had me thinking this was "F2P" crap - and I didn't pay it much attention for quite a while. Later I started to get quests to complete a certain amount of rushes, so I had to try it out. I found that it is not as bad as I first expected. Rushing gives pretty good odds of getting resources a little quicker, unless you do it over and over on the same room (difficulty seems to increase by roughly 10% per attempt, with some long cool down). You can make your odds a little better by using Dwellers with high Luck, and/or waiting till there isn't much time left. Some tips I'd give when doing quests are that it seems easier to rush rooms that are smaller, and it is always a good idea to have the room at its Dweller capacity to deal with any incidents. Since Radroaches are one of the two things you'll run into, make sure your room is well armed.
7. More sexism
In what was probably a smart move for various reasons, Bethesda decided to add a mechanic to the game where pregnant female Dwellers run from any conflict and are not available to enter the Wasteland. However, given that you are greatly incented to keep female Dwellers pregnant, this makes them second class citizens when it comes to gear and especially weapons. Its hard to justify giving them your nicest guns when you know they are just going to run away if a fight breaks out. So, while it pains me to say it - give your best gear to Dweller dudes first, because you can be sure they will always be available to use it.
8. Obey the pulsing gear
Without much pomp or circumstance, there will be times when your Vault reaches its population limit (usually with a host of pregnant women wobbling around with sore feet and aching backs). At such times, the gear on the upper left of the screen will start pulsing to indicate your vault is full. At this point you need to expand a Living Quarters, or upgrade it, to make more space. It's an easy fix, but remember as mentioned above not to add rooms too fast without also adding Power capacity.