Should I use an adapter or converter for my travels to Europe?

For a hair curler, it reads 120-240V~50/60Hz,40W. I will also be charging an iPhone & apple watch as well when traveling to Europe

The hair curler is designed to operate at all voltages that you will encounter so all you will need are adapters to suit the plugs in use in each country. IPhones and Apple watches (and pretty well any other digital device like a notebook) also work on all voltages so again, all you will need are appropriate plug adapters.

If it reads 120 - 240 V then you need an adapter.

If it reads 120 V then you need a step down converter. Europe uses 220 V so if you plug a 110 V electronic into a 220 V adapter, you'll fry it.

I would play it safe and plug everything into a step down converter. It's more expensive but better safe than sorry in case you make a mistake.

Unless the appliance states only 120v or the plug mentions only USA voltage, i.e. Not a range of 120-240v as the input range, you'll have to assume it will only take 120v when plugged directly to power sockets, in these cases you'll need a converter, which can be $50-100 each, plug converts are just a few $s and are all you need with appliances/chargers that have the voltage range displayed.

Why do you need to curl your hair?

The curler is fine to operate at any voltage
for the iphone you will need to buy a voltage adapter/converter
that converts Europe 220V to US 120V