Start from an early age.
Make it a race / game with music on the background.
We always did tidy up time together before dinner, no dinner on the table before things were tidied away; it worked.

I would try to make it a game, worked for me when i was a kid

When my daughter was growing up, we made tidying up just part of the regular routine. I encouraged her to put things away before starting anything new. Now that she is older, it’s just something she automatically does.

Not sure what’s best … whether to go with bribery, or whether to say “you can’t do … until everything is tidied up.”

We play games usually. “Can you put all of the red blocks back into the bag? What about the blue blocks?” If playing with a puzzle or something similar, we don’t allow another toy or activity to be pulled out until that is packed away and we explain to him that the puzzle will get mixed up if other toys are out and then we won’t be able to do it anymore. If all else fails I just let him know that I will help him pack away if he does it too.

Shout, yell, cry, jump up and down, bribery, confiscate when they don’t put things away, reasonably discuss it with them

once every two weeks we get out a broom and sweep everything into a pile in the main living room, yes it is huge, but if they do not put it away we put it in the bin..(yes we do retrieve the expensive stuff but only after the kids think they have lost it for ever and dont bring it out till after the next big clean up… )The kids soon learn not to let their stuff go in the bin.. Now only the junky rubbish gets thrown in the bin.

Yes that sounds like a good plan and taking the lead to show the children how it’s done repeatedly is effective.

When my daughter was young I used to clean up with her. I can’t stand mess so I was sitting with her and slowly everything went back to its place. Because of doing that regularly, it became a habit for my daughter and, with time, she did everything by herself.
I think that a little time spent now with your kids cleaning up, will help a lot in the long run.
Success!!