16 yo, freaking out about money being taken from my account

You've almost certainly signed up for something where you thought you were making a one-off purchase (probably the £3.50 transaction) but somewhere in the small print it said you were committing to a regular subscription with cancellation terms etc. If I'm right, this isn't a pure scam as such but certainly sketchy business practices.

Do you have any records (email etc) that would tell you what you might have bought for £3.50 on 8th March? That would give you a clue, and hopefully a link to how to cancel, although be aware that those types of sketchy company often make the cancellation process deliberately difficult.

If you aren't able to contact the company to cancel, then contact your bank and tell them you withdraw the authorisation for this ongoing charge. They will cancel it from their end (without you needing to get a new card or anything). The worst that will then happen is that the company you signed up with will contact you claiming you still owe them money - and at least then you'll have a way to contact them back to say "I no longer require this subscription".

For most folks your age, this may fill you with existential dread, but the best thing to do is to call the number on your card and talk to a human being about the situation. Call the number printed on the card, and not some other number.

Be prepared to answer some authentication questions (have the most recent statement available) and it's better if you call from the phone number associated with your account.

Guy working in credit card processing here. Let's go over some of your concerns.

Why are they asking for the first 6 digits of my card? Don't most receipts only ask for 4 digits? If I give them the first 6, can they steal my information?

First off, they already have your credit card info. If they didn't have it they wouldn't be able to bill you.

Second, if they were trying to steal your credit card, they'd be asking for everything (full card num, CVV, Billing address).

Let's talk about credit card numbers. The first 6 to 8 digits of a card are known as the Bank Identification Number (BIN). This is used to identify the bank that issued you the card (also known as the issuer). While most receipts only print the last 4 digits, back-end reporting on the merchant side will usually have those first 6 digits as well. If a merchant has run more than 303 different cards, the probability of two of those card sharing the last 4 digits is over 99%. Using the first 6 digits helps narrow it down significantly.

Couldn't they still brute force those other 6 digits?

Yes, but it's not very efficient.

At a minimum to run a credit card you need at least the full card number and the expiration date. Using a Luhn Check, you could narrow it down to around 100,000 cards numbers. Cards typically expire within 3 - 5 years of the time they are issued so that brings it up to 3,600,000. And that's assuming the merchant isn't requiring CVV and AVS. It's faster and more reliable to get stolen card lists online than it is to test randomly.

What should I do? I don't recognize this service.

Reach out to the merchant first if you can contact them. As many of the other answers have said, ask for any contracts you may have signed. Figure out if/why you signed up for it. Try to get a refund from the merchant first before going to your bank.

I determined I didn't sign up for this and the merchant isn't responding/working with me/honoring their contact. Now what?

File a chargeback with your issuing bank. They may issue you a new card depending on whether or not it the issuer thinks the card was stolen.

The merchant will be required to return the funds to you. The merchant may dispute the chargeback claiming the charge is legitimate (as is their right). In card-not-present environments (e.g. buying stuff online), the bank will almost always side with the cardholder (this is you). However, a merchant can still decide to take legal action to get their funds back even if they lose their dispute.

Is my issuing bank going to give my new card number to the merchant or other businesses that have my card on file?

No.

Edit: As @Syfer Polski has pointed out in the comments, it is possible for the merchant to get an updated card number if they sign up for an account updater service. Not all processors support this and not all merchant use this so the merchant may or may not get your new card number.

What do I do?

Since you can't identify the service, just contact your bank immediately, tell them those are fraudulent transactions, they will cancel/replace your card and more than likely refund you those charges.

In the UK, you call the fraud department of your bank. Have your bank statements ready. You tell them which payments were made which you didn’t authorise. They will put the money back in your account, block whoever did this from taking money, and they may decide to send you a new card.

(That’s what Santander did for me, and they told me that any UK bank they are aware of will do the same thing).

PS. There are plenty of companies tricking you into signing something that may look like authorisation. It’s not authorised - according to my bank - if you are not aware of it.

PPS. Being sixteen years old even if there was a valid contract he could void it for the next two years.