You're in good company, as others have said, most toddlers go through this type of stage. I'm right in the thick of it with my 21 month old daughter. It's all a bit of trial and error, and some distractionary techniques have a pretty short shelf life, but here's how we generally make it through a trip to the grocery store or mall:
-I make sure to bring a sense of humour, something is going to go wrong, be forgotten or get difficult (I also have a 4 week old son). Don't worry about people staring or commenting, for the most part people likely remember what shopping with a toddler was like. I find if I stay calm and try to laugh to myself at the craziness of the situation, then I'm less likely to get completely stressed and feed my daughter's tantrum. If I start to get angry or frantic, then she is less likely to calm down. If I quietly ask her to look at me she'll eventually stop the howling.
-Have an army of supplies. In the grocery store, I don't even entertain the idea of letting my daughter out, in fact I do that safety belt up as tight as I can get it. I bring sippy cups, juice boxes, cheese string, goldfish, a mini magnadoodle, her toy purse, an old cell phone, a pen and paper, stickers, just anything that might grab her attention for a few minutes.
-Engage her in the trip. We walk around the store and I'll talk about what's next on my list. As in, oh look red apples, we need 5 apples, let's count the apples. Can you see the bananas? Let's go find the bananas! I also give her a few things to hold, like a box of pasta.
-Have a speedy exit plan. I write a list of things we absolutely need and then have a full list of all the things I want to get. If a meltdown starts and doesn't look like there's anyway it's going to end, then I quickly grab the things I really need and either come back the next day, or that evening once my husband it home.
-If at all possible, avoid shopping with a tired or hungry kiddo.
At the mall we use a toddler harness, it's got a monkey that goes on her back and we got her used to it by putting it on at home, calling it her monkey backpack and letting her play with it. She now asks to wear it whenever we get out of the car. I too don't like the idea of a harness on a child, but what I like even worse is running all over hell's half acre after my child as she darts toward an escaltor!
Anyway, good luck, it's a trying age. I try to remember that she's just figuring it all out too and there will be days that go great (and I make sure to let her know what a fantastic job she did at listening or walking or sitting or whatever) and days that aren't so great...and those days are why God created wine. ;)