Hi L.,
I feel your pain. If I knew the answer to this question, I'd be a rich woman. I guess to best correct the behavior you have to understand it. Obviously you have to shop, etc. Can you identify any contributing triggers such as hunger or lack of sleep? Within reason we have to tolerate a little bit of temper tantrum from a child of this age and some children are more prone to temper tantrums then others. Of course they need to eventually learn self control. This behavior is socially unacceptable.
A child of 19 months to about age 3 is really too young to reason with and usually is acting out on a feeling of frustration and lack of control.
Most importantly we don't want to reinforce the behavior or lose control ourselves. (easier said than done) There are somethings you simply cannot compromise on and your child needs to understand that. A little work now will yield a whole lot of benefit years down the road.
There is a battle of wills of sorts over who is going to be in control. You cannot lose this battle. If your child wants something he cannot have tell him once. (make sure you have his attention) hold his face in your hands and tell him... "Not now, mommy has to ....." If he proceeds to a meltdown. Say nothing and try to wait it out. If it passes proceed. (ignore the stares of onlookers) If the tantrum does not pass, quietly pick up your child and leave. Even if you must hand a cart of groceries over to an employee with apologies.
I understand what a hardship this is, but you cannot allow a tantrum to go on indefinately and you CANNOT give in to your toddlers demands.
Of course, it is easier if you can avoid this scenario altogether. Make sure your child is rested, fed and toileted before venturing out and then give ample warning about the shift in gear. If he is playing tell him "we will be leaving shortly to go shopping, can mommy help you finish that game?", You could also ask him to pick out a favorite stuffed friend to go with you. If your child feels he is an active participant of the plan he will be more likely to cooperate. You could allow him to help pick a cereal etc. Be sure to praise his cooperation.
Parenting a toddler can be very frustrating. Understand your previous goals of what can get done in one afternoon have to be restructured to your new life. And don't forget to enjoy every moment. They really do go by fast.
God Bless You,
J. L.