How do you get them to sit in the seat and stay there?
Distraction and boundaries. A mixture (according to your own parenting style) of treats/activities/toys and just setting boundaries of what's acceptable and what isn't.
Things like shopping, unfortunately, will no longer be a simple affair of trundling around with a shopping trolley unless you go childless. For a long, long time to come.
Talking, singing. Things like grapes, raisins, lollipops. Giving baby something from the shopping basket to hold (and keep swapping it). This alongside constantly putting back into seat and being stern about saying no. Not positively re-enforcing bad behaviour. Repeating and modelling expected behaviour.
Once older bribery works - behave and you can have a magazine/toy/sweet. Rewarding good behaviour. Involving the children with shopping - giving them their own list and having them help choose what we buy.
Also when older and into the refusal tantrum years I have been known to plonk child on floor in supermarket facing the wall until ready to do as told, sit still in the trolley and behave.
Better this than be one of those Mum with a screaming out of control, badly behaved child in the supermarket 