I've just bought a Giant ATX 2017 and it's my first bike in 20 years, so I'm having to relearn everything again. I took it out for just the second time today and, having decided to experiment moving out of granny gears for the first time (I live in a very hilly area), I realised I have an issue.

It's a 3-8 setup and going from 1 to 4 is fine, but when I go to 2-4, it doesn't quite catch and makes a grinding noise, only fixed by moving back down to 1-4 again. I got it up to 3-6 but the chain came off due to the extreme jump. I managed to dismount, roll it back on and take off again but as soon as I downshifted to 2-4 it was the same issue - it wouldn't engage properly until I went back down to 1-4.

I'm a total noob when it comes to this so does anyone have any idea what the issue could be? And apologies if my explanation is unclear - I'm new to the technical terminology.

New bike, get the LBS you got it from to fix it, sounds like it was never set up correctly during the pre delivery inspection

The cables typically settle in some during the first bit of riding. Needing an adjustment shortly after purchase is normal.
Take it back to the shop, they should adjust it for free.

Agreed, but not after 2 rides, 3 months sounds more normal for this adjustment.

Unfortunately my LBS isn't a Giant dealer. I tried it out on a trip to London, 55 miles from where I live and it was supposed to have been tested before being delivered to me.

Could it be something as simple as the the chain needing lube or cable tension adjustment that I could do myself? I could take it to my LBS but it's two miles away and I don't drive.

Try the Park Tool website, and there are plenty of how to videos on Youtube (try GCN to start with), alternately, time to make friends with a LBS, may be worth it for shop rides /socializing if they offer this. (for taking it to a LBS 2 miles away, your not going to kill the bike riding with skipping gears, and it's not far to walk back home after if they keep it to fix)

The adjustments are very standard and fairly easy. It does sounds like it's pretty far off though. Has it always been flaky? If so you might want to complain to the shop you purchased it from.

Any bike shop will be able to adjust it, there's nothing Giant specific required.

Yeah, I've seen that site, it looks really useful and there are a ton of great great videos out there too. Truth is, as a complete amateur, I'm reluctant to do anything myself so I think I will just take it to my local LBS. I'm sure it'll be a simple fix and I need to buy some accessories anyway.

Turn the barrel adjuster on your left shifter counterclockwise 2-4 turns.

Then turn the high limit screw (the one labeled with an "H") on the front derailleur right (clockwise) until the chain stops dropping off the outside when you shift. (Yes, trial and error.) (Could be anywhere from an 1/8th of a turn to many turns depending on how far out of adjustment it was. If the bike stops shifting to the biggest ring, you've turned it too far.)

They will still help, be nice buy something from them for helping you figure it out..

You got a taxi? a Library? Bike repair book to check out ? (to learn DIY, I read books, before the internet)

Follow the Park procedure, and make adjustments in small increments.

Cheers for the advice everyone. I contacted the shop I got it from and they said they'd pay for any repairs if I just email them a photo of the receipt, so I'm happy with that.

There is nothing unique or unusual about Giant drivetrains. Any LBS should be able to make the adjustments you need.