Is it better to build new rooms or upgrade rooms in Fallout Shelter?

This actually depends on how you play the game, and how many dwellers you can equip.

Play Style

If you only play for a short period, and usually have at least 20 minutes or so between play periods, then getting more rooms is not a bad idea.

This is because you can place only one dweller in the room, and it will still be ready by the next time you play. During the interim, your dwellers do not consume resources.

If you play for longer periods or more frequently, then you care about the cycle time for each room. If you play for 10 minutes, and your food room with one dweller takes 18 to finish, then you're going to have starving dwellers.

So, in the case where you play more often, you need to sufficiently staff your room, making it more difficult to expand. Additionally, upgrading causes the room to produce more food per cycle, increasing the food-per-second.

Either way, you should be careful not to fully upgrade your rooms unless you have tough dwellers with good weapons, or else the incidents will be a serious problem.

Dwellers

More rooms means either more dwellers, or less dwellers per room. I talked about the number of dwellers per room above.

If you plan to fully staff these new rooms, you'll need a lot more dwellers. You want everybody in the vault to have a decent weapon to deal with incidents, so you shouldn't expand faster than you're increasing your weapons supply. If you have plenty of weapons, then creating and fully staffing new rooms might be a viable option. Don't create new rooms and staff them with six level one dwellers carrying BB guns.

What I Do

I usually don't play for more than 5-10 minutes at a time, and only look at the vault a few times a day.

I build one each of food, water, stimpacks, and radaway production. I try to fully staff the rooms as quickly as possible. At some point, I will upgrade each room once, but not twice. This is determined by need. If I'm running out of water, I upgrade the water room.

Because I then build training rooms, and am constantly upgrading my dwellers' outfits, these individual rooms are enough for quite some time. I eventually build second food and water rooms when the single rooms are no longer sufficient to keep everyone fed and watered.

Clearly, there are a lot of factors involved that will affect your individual vault, but I wouldn't expect additional food and water rooms to be needed until you have around 50 dwellers.

I currently have a survival vault with 71 dwellers that just recently built new food and water rooms, and a survival vault with 37 dwellers that still only has one of each.