Karlie here from the Shopify Support Team! If you're looking to create a logo for your store, then I would recommend starting with our two blog articles; How to Design a Memorable Logo in 7 Steps (We Make One From Scratch), and 5 Easy to Use Online Logo Makers to Design Your Brand. These articles will help you figure out what type of logo will fit for your business, how to create a logo and where to use your logo.
When you're ready to create your logo, there are a couple of different logo generators that you can look into. I would recommend checking out Hatchful or the Oberlo Free Logo Maker. Both of these generators can help you create a logo that will fit your business needs.
I hope this has answered your question. If not, or if you have any further questions please let me know. I will be happy to help!
The best program to create a logo that is unique and not like the others is Adobe Illustrator. If you have the patients to learn it'll be good for you. You pick up a skill and learn how to design a logo. Plenty of tutorials.
There are free logo creators online but most are not unique, and being you have one in mind it might not be what you're looking for.
If you can't create one, send me an email