My DS is 2 1/2, I spent the last year or so NOT taking him anywhere if it was avoidable. He stayed home with dad, or with my MIL who drives down once every couple weeks to spend the day with him. I've started taking him out more recently and it's all about the planning.
**Make sure DS is rested, fed, and has had your undivided attention for a little bit.
**Trips need to be short---have a shopping list. Roll through the store and get what's on your list and head out.
**Browsing needs to be at a minimum. If I need an item and I am not sure where it is...I look quickly, and then come back without kiddo if I couldn't find it. This also means, no trips to the mall to "just look". A 2 yr old attention span can't handle that kind of outting.
**Distraction really can be powerful. I went to rent a movie recently, and had it all picked out online (no browsing). DS is scared of strangers and refused to move beyond 10 steps in the door. Luckily, there was a case for Dumbo the movie nearby. "Look DS, elephant!" I handed it to him and he was distracted long enough for me to grab the movie I wanted.
**Bring his favorite cup, book, toy. If you have to sit for a while bring crayons or a puzzle. I used to bring a couple options and bring the next one out when the previous "fun" wore off...always save the best for last.
**Involve him. "Can you hold mommy's list?" "Should we get bananas today? Which ones look the best?" I often can be found tooling around the store making zooming noises and funny faces at DS, oblivious to the people around me.
**When all else fails, ignore the tantrum and stay very very calm. He will feed off your nervous energy. Grab the essentials and get out. If you can't stay calm, abandon the basket and head home. It isn't worth getting all worked up over. Come back alone after daddy gets home.
These are the things I find help with my DS. Hopefully, you'll find a plan that works for yours. Good luck!
C., Battle Creek