DOMS - when to go back to the gym?

You could try to go today but if you're still quite sore, it's likely going to affect your form so it might not be a great idea. Wait another day, do some more gentle cardio and stretching. How about some upper body work?

Work through it. Once you're warmed up, the pain isn't as bad, sometimes barely noticeable.

The general rule is that if the muscle is sore to the touch, you should *NOT* work it out. If not, then it's safe to work it; however, you might consider doing a workout of other body parts which aren't already sore.

get to the gym now! I always plan for a cardio day after leg day, the sooner the better, and do some low intensity steady state cardio, keeping circulation to the muscles will aid in the rebuilding process and help to reduce inflammation. I would not do any kind of strength traing on those muscles until the pain has subsided. It does get better.

Lastly, foam roll your quads and hamstrings. Doesnt take long maybe a minute or two on each leg but it helps. I foam roll after the cardio. I hope this helps!!!

Also, if you can up your protein intake for the 24 hours after strength training session. That has seemed to work for me. Not saying that it actually helps but if your mind thinks it helps that is good enough for me. The placebo effect has been proven countless times.

The sooner you get some blood back into the muscles the better.

Also, invest in a foam roller and use it after your lifting sessions, should help ease the DOMS a bit.

If you're *this* sore, perhaps you tried to go too hard too soon in the previous workout? Adjust your expectations, avoid an 'all or nothing' mentality and realize this whole thing is a lifelong journey.
What lifting program are you following?

So I lifted for the first time on Monday night. The last 2 days have been painful.

It's started to ease a little today - my thighs no longer feel like they may explode with every step I take - and I can now slowly and painfully drag myself up 2 flights of stairs at a time, but sitting down and walking down the stairs is still a big problem.

I've been keeping as active as I can - I've managed to still do my 10k steps - albeit very slowly for the last couple of days. And I've managed a gently swim and an hour of pilates.

My brain tells me I want to go back and do session number 2 now - my legs aren't so keen.

So do I take a couple of painkillers and put my legs through more squatting - or do I give myself a bit longer?

What you're describing is exactly how my legs felt after the first couple times I squatted. I recommend warming up on the rowing machine before you lift, and then doing some dynamic stretching (plie squat, lunges) then foam rolling the quads and glutes and then lifting as normal. Eventually it won't be so bad if you stick to a regimen. I'd say after about 1 - 2 weeks of consistent lifting you won't have this level of soreness.