Hi,
I understood that you are looking for good shopping malls and the cheapest shipping companies in Dubai.
I am residing in Dubai for the past few years and I visited most of the malls here. According to me, I would recommend you to visit Etihad and AlBarsha mall in Dubai which comes under Union Coop, the largest consumer cooperative in the UAE.
I spent most of the weekends with my family there. You will get everything you need for your home here. There is a separate entertainment zone in both of the malls where you can enjoy with your kids.
You can do all kinds of shopping such as baby products, jewellers, cosmetics, electronics, gifts, and so on. The food courts, cafes and restaurants are also the main eye attractions of these malls. So I would recommend these two malls in Dubai from my personal experience.
For your reference, you can check here for getting an overview
Etihad mall: https://etihadmall.ae/en/
Al Barsha Mall: https://albarshamall.ae/en/
Regarding cheapest shipping companies if you are referring to online delivery services, Union coop has a hypermarket, where you will get all the items including groceries delivered to your doorstep by their own delivery service.
Or else, please refer this link to know the major shipping companies in Dubai: https://aiwa.ae/category/shipping-lines-agents
Hope this helps, Thank You.