As far as I know, the only airline where you have to check in online (even if you are going to the desk with a suitcase) is Ryanair. And if you book certain packages they include airport check-in. In any case you will get loads of e-mails before you go. When you have checked in online, put all of the boarding cards onto two different phones so you have a backup. Or print them out, if you have a printer, because it's less fiddling at the airport compared to handing over your phone while controlling small people.
But since you will presumably be checking in luggage, online check-in won't save you any time at the airport. I would try to arrive at least 2 hours early in high summer, and depending on the airport (cough Stansted cough) I might consider buying priority security. Of course if you do that there will be no queue (Sod's Law, innit), but if you don't and there is a huge wait it will be horrendous, both in terms of time and hassle with kids. (You don't need priority security at Palma for the return, as security is always very quick there and they even have special lanes for families with children.)
Regarding cards, you can use them almost everywhere. Check your bank's overseas/Euro zone usage conditions for what they charge. It's come down quite a bit because of competition from Monzo and co. Hopefully there will be no minimum fee and they will only charge you about 2% on top of the official exchange rate. If the card machine in the shop/restaurant offers you a choice between paying in Euros or pounds, always choose Euros.
If your passports are not blue (actually I think they're black), check their validity. They need to be less than 10 years old (that is, from the day of issue) on the day you travel, and to still have 3 months to go until the expiry date on the day of your return flight.
Can I ask where you're going in Majorca?