The biggest problem would be magnets. Iron has the ability to keep magnetic fields contained in a much smaller volume. An air core transformer is much larger. Motors too would be very large. Other materials are magnetic, but not nearly so much as iron. Alloys? Maybe.
C.M. Kornbluth wrote a story, "That Share of Glory" about a planet that had little or no transition metals on it. (Copper, Iron, Nickle, ....) Their answer was ceramics. Given how well our Corelle dishes wear, this has a possibility.
Another short story was rebuilding after an nuclear war. All the easy ore deposits were used up. What to rebuild with? Their answer was a form of tempered glass.
Not having steel would be a huge handicap. The replacement of wooden hulled ships would be much later. How would you railroad? Sure, bronze. But the price of bronze compared to the price of steel would mean that you would use it sparingly. Do a price comparison on a bronze hammer compared to a steel one.
Aluminum is a good replacement. Aluminum Magnesium alloys are even better. And it's cheap enough in bulk for ship building. The MHS Sheffield was an aluminum alloy frigate(?) in the Falklands war. Got hit by a single exocet missile and burned. I didn't realize that aluminum could sustain a flame, much like magnesium.
Titanium is another expensive replacement. When external frame packs were still in when I was a boy, you could get a steel pack frame for 20 bucks, and aluminum one that was about half the weight for about 35, and a titanium frame that took another 4 oz. off it for $300.
Getting to the Aluminum Age from the Bronze age would be hard. Cheap steel made the industrial revolution possible. Without iron, you'd be stuck with bronze/brass. https://en.wikipedia.org/wiki/Bronze
This page: http://www.roymech.co.uk/Useful_Tables/Matter/Costs.html gives typical comparison costs. This gives you a way to look at alternatives, but with no iron the demand for everything else skyrockets. For story purposes I would take the multiplier and do a further multiplication by a factor of 5. So for example, a steam locomotive made of bronze would be a factor of 6 for bronze, and a factor of 5 because no one has steel, so 30 times the price.
Of course if copper, zinc, tin are more common due to iron's absence, this will affect these factors.
Once you get to the Aluminum age, shortages disappear. Aluminum is very common. It's just expensive to refine. At this point, you get down to things that are only a factor of 4 or so more expensive. I think your industrial age will limp along until the aluminum industry gets going.
Look at bicycle technology. Steel alloy frames are in the 30 pound mark, and you can still get a serviceable $150 buck bike. With carbon fiber you can get the weight down below 12 pounds, and a price tag comparable to a good used car. Note however that the chain and gears and spokes are still steel. There are bronze alloys that would work for this, but at a price factor of 6 to 30 times as much.