What's the cheapest way of paying by card abroad?

There isn’t a general solution, because what’s best for you depends on how much money is involved and how often you need this. For casual tourists, charging a few nights in hotels etc, it’s not really worth the hassle to search for the absolute best deal.

For somebody who travels a lot within the Eurozone and needs to convert £10k/year or more, it is worth setting up a bank account in the Eurozone (but be sure to find one that issues a debit card and charges no fees for maintaining it!) and then transferring money into it via Moneycorp, which will give you a much better exchange rate than your bank. Doing all this will take you several hours of work and you end up saving 2% of the amounts at most.

You can find a bank that allows you to use your card with accounts in multiple currencies. For example, my main debit card is connected to a CZK account and to an EUR account. When withdrawing or paying, the appropriate account is used and there are no additional fees by the bank. It is then up to you how you get the money there, you do not have to use the bank’s conversion services, but you can use any other conversion services with better rates or you can use your direct sources of income in the other currency, if you have any.

Note that ATM fees may still be charged by the ATM company. For example, ATMs in Austria this week charged us between 2 to 6 EUR and directly send the request for the amount being withdrawn increased by the fee to our banks.

This is a much discussed topic at my place of work (in the UK), which involves a lot of European travel. The general consensus is to get a zero fee credit card which operates on the card scheme rates (Visa or Mastercard). I personally use a Halifax Clarity credit card.

Its important to note that while there may be options that are slightly better in financial terms, these are very marginal and often come with other drawbacks (eg customer service, ease of use, hidden costs as you have discovered already etc). Unless you will be spending a lot of money the margins might not outweigh the convenience of using a standard credit card from a reputable UK bank. My sales manager has a specialist credit card, which comes with a hefty annual fee, but it’s worth it for him as he puts over £10k per month through it.

In my opinion a no fee credit card is the best compromise for the general traveller/tourist, but if you travel a lot or expect to spend big then its probably worth something more specialist.

It’s also worth checking the advice over at https://www.moneysavingexpert.com/travel/cheap-travel-money/

Wise is a very simple, much safer solution. None of the above applies: it’s a public company with all the transparency that comes with that, profitable since 2017. It provides you with local accounts and a debit card which draws against them. The fees for withdrawing cash certainly tells you they don’t want you to do that, but payment wise it’s one of the best or the best card currently there is. Total transparency on fees when spending in any currency, of course. You can have full accounts in EUR, USD, CAD, GBP, AUD, HUF, RON, NZD, SGD, TRY — these are all accounts anyone (including yourself) can send money to. (There are many more currencies you can hold money in but those are just entries in the ledger of Wise).

The lowest cost way that I know of involves two things:

1. Get a debit card that reimburses for all ATM fees, and does not charge a conversion fee. For example, my card is from Charles Schwab bank (USA), and the conversion rate is the official Visa rate exactly.  At the end of each month, they refund any ATM fees I paid during the month.  (Mastercard rate slightly lower.)
2. When paying by card, make sure the option to pay in local currency is selected.  Don’t let the merchant select your currency, which lets them and/or their bank benefit from the spread on the conversion.

A slightly cheaper option¹ is "Wise" (formerly TransferWise). Last time I checked, that would have saved me two US pennies on a thousand dollars.  But it’s not practical for other than long visits, because it requires a local bank account to transfer into (as I understand—I haven’t actually used them for a transfer).

¹No longer cheaper—see comments.

This made me think I should have just loaded GBP onto my Revolut card instead, as my UK bank account would not have charged for moving money to another UK account. Then, when abroad, I could have still opted to pay in EUR, and there would not have been a conversion fee. Is my understanding correct?

Yes, exactly, that’s the best way to use Revolut or any other similar card to avoid fees (with the caveats mentioned by Frank regarding Revolut, like the extra margin on week-end payments and a handful of currencies, see also https://www.revolut.com/legal/standard-fees/). You can or could also keep Revolut accounts in several currencies and transfer funds between them but I think there is a monthly cap on those transfers after which fees do apply. Cash withdrawal are also strictly limited.

I know my UK bank account does charge a fee for conversions abroard, which is why I would like to get a different card to take with me. I have looked at other options like Monzo which do not charge for conversions, but I am still confused.

Should I keep GBP on the card, and when abroad pay in local currency? Or, should I always ‘load’ local currency on my card before I go?

I am not familiar with Monzo specifically but with other similar offers (both fintechs / neobanks and some more traditional banks in the EU) and in general you are better off keeping the money in whatever currency you received it, avoid getting cash, and pay with a low or no-fee debit or credit card whenever you need the money.

You also want to avoid any conversion from the ATM or merchant if offered. Very often terminals abroad will do that since your card is associated with a GBP account (see Why are two currency options available when paying via card at airports abroad?). Instead, always ask to get charged in the local currency. Similarly to what happened with your initial GBP-EUR transfer, if a merchant charges you in GBP, you will incur (often excessive) fees and that’s completely outside the control of your card issuer.

Same things for billing in a foreign currency from hotels or other businesses, those are almost always less attractive than letting Revolut or similar services handle the conversion.

Revolut tends to be the cheapest solution when paying by card when traveling, as it uses the current market currency exchange rate, but it does have fees in some cases:

• for some currency exchanges
• for all currency exchanges during the weekend
• ATMs if withdrawing more than some small amount monthly (currently only 400 euros can be withdrawn monthly without revolut fees with a premium account). Also it doesn’t help avoid the ATM operator’s fees for some ATMs.
• the currency exchanges without fees have a monthly limit if not paying for a premium account
• etc

Also, their customer support can sometimes be a pain and they froze my account for no reason so don’t solely rely on them.

See https://www.revolut.com/en-US/currency-converter; warnings: some fees such as the weekend 1% fee are only visible in the Revolut app.

Regarding other banks: many banks use the Visa exchange rate or the Mastercard exchange rate, which less favorable that the actual currency exchange rate. From https://nomadgate.com/visa-vs-mastercard-exchange-rate/:

Visa is on average 0.30% more expensive than the mid-market rate, while Mastercard averages 0.21% above mid-market.