Is there a bag I can use to safely carry my passport and ID whilst travelling?

There are wallet-like holders specifically designed to keep your passport and some other cards (credit card, healthcare card etc)

They look like this and can be bought in just about any airport shop I've been to anywhere in the world.

This thing is though you can't keep anything on you, everything you're carrying on the plane has to go in those plastic trays through the X-Ray machine. The X-Ray won't damage your passport and no one would try to take your passport, there are so many security people there it would be silly to even try.

That said if you are still concerned you can hand it to the security person and they'll give it back right after you go through the metal detector, this way you know for sure nothing will happen to it.

No need to worry though, everyone does this all the time, in all the security lines I've been in, never seen anyone have problems with that.

Hey Sarah here is one possible solution,

Keep your passport IN YOUR HAND when you go through the security check. Note that, anyway, in some/many locations they make you do this.

Your bag goes through, and you walk through the scanner with your pport/boarding pass, which you show to the scanning-person.

Regarding generally where to keep your passport:

KEEP IT WITH YOUR WALLET.

It's that simple.

Wherever you keep your wallet (I keep mine .... "in my pocket"), then keep your passport in the same place, with it or perhaps even just inside the wallet.

That's one very simple solution that keeps it clear in your head.

You don't often lose your wallet ... right? Your passport is no more or no less important, they're equally important and critical. So, keep one with the other!

It will always be clear in your mind!

In some cases you MUST put your wallet, and passport, and phone, in a plastic tray and have them scanned. If that's the case ........ you must do that.

At airports where that is the case, the "BERG" underwear thing will not help you at all in any way, unfortunately. (Indeed, it will conveniently direct thieves to the fact you are wearing one. :) )

For international trips, I use an wallet that hangs inside my trousers on a belt-loop. The fabric is extra-reinforced. This isn't my brand, but it's the same idea. My wife prefers the money pouch (similar to this). Travelling together I usually give her my passport too, and we watch very carefully for the tray containing the passports to come through the scanner.

You should try to relax, traveling is a fun experience and although it may seem very complicated once you understand the process its is very simple to get used to it.

You will need to put any kind of bag into the machine. No matter what it contains. So if you put your passport into this thin bag, you'll have to take it off to put it through the machine.

If you are carrying a laptop, ipad or tablet with you, you'll need to take it out and place it separately.

Always, always, always carry your passport with you on your person. You can walk through the x-ray machine with the passport in your hand without a problem and no one will stop you for carrying the passport. Similarly, make sure you carry your ticket/boarding pass with you as well at all times.

The idea of a passport wallet like what was posted by @blackbird57 is an excellent idea. Another variant (and the one I personally carry) is this item:

Since it allows me to carry a few other things and also fits my mobile phone into one item.