One of the challenges I tend to face on and off is going to the store... any store. If you think about a trip to the store there are a lot of things that can trigger your child's sensory issues from people, lights, sounds, stuff, and sometimes even how you keep your child with you such as the cart or a harness.
Hopefully some of these might help you cope through the store.
#1 - Prepare your child. Let them know in advance you are going to the store. Inform them what it will be like and how they are to behave in the store.
#2 - Make sure you bring with you some supplies to help your child cope. Snacks, drinks, small toys, a comfort item, and so forth can make your trip a lot easier. We ended up getting a small fanny pack quite some time ago for this purpose and sometimes we use a backpack depending on our son's needs.
#3 - Make a list and follow it. While this isn't always easy it is amazing to see how my son does when I tell him on a sensory type day that we are going in for 3 things and list them off. Each item I get I tell him what is next on the list until I am done with my list. This helps him know that we aren't going to be out forever. This can also be done with the stores or errands you have to do. I usually got straight home afterwards.
The other thing I have found that I have to do constantly is think outside the box. My child is sensory seeking and so I tend to think in ways he can have various sensations. For instance, at Walmart I found something called a Fusion Ball. It is one of the most annoying toys I have found as it is a ball and it vibrates all over and makes noisy music that you can't turn the volume down on. It is $5 and it was well worth it although it didn't last more than a couple weeks as the outer cage broke. (I have mixed feelings about that.) My son would bring the ball with him and hug it to himself really close as we would go through the store as it jiggled him around. People I am sure were sick of the noise but it kept my son under control.
What can you do to help your child? If lighting bothers your child then what about getting them a pair of sun glasses? Find something that works for them regardless of your opinion on how they look on your child. If sound bothers your child what about some thick headphones? The nice thing about headphones is that they can be connected to some device that can play songs or tell stories. If they are overwhelmed by people then maybe bringing in a blanket for them to cover their heads and bodies up might do the trick.
The truth of the matter is we have to go to stores and while many times not bringing our SPD child is easier it isn't always practical. Maybe you need to make a sensory kit for when you go out to fit your child's needs that is strictly for going out in public. There are many small toys and gagets that you can make as well as tiny toys that you can get from a party store such as Party City.
Most of all, make sure you compliment your child. Going out in public tends to be a very stressful thing. I have been told that it can affect your child not necessary immediately but show up a few days later. I have yet to be able to identify this with my own son. I have noticed a lot of praise has worked wonders especially in the small things that he's able to do. It doesn't happen all them time. In fact, his abilities vary day by day but with each day I am reminded that he's learning tools to cope in the world he will soon face on his own. Try to find one thing... even a small thing that they did right whether it was respond one time appropriately to a command or remark you made. If it even means they held the cart for 1 minute. Look for those little opportunities to compliment them espeically on a hard day.