My background. I have developed stores on osCommerce, ZenCart, Demandware and Magento. I have also helped people get running on Shopify as well. I see your options as 3 fold: hosted solution with a custom theme, an open source solution with custom theme, totally custom site. I think you can rule the last option out if you don't have a unique spin on shopping (ie, you are not listing products, clicking add to cart, checking out).
If you like Shopify and just want a better theme, I would second the others that have advocated getting a designer and developer to make a custom skin for you.
If you don't like Shopify or need more then I would recommend UpShot commerce. Their pricing is reasonable and they have a really solid team. They bring a lot to the table and have good developers to help you if need a customization here or there.
Self Hosted / Open Source
Lots of people recommend Magento, but I cannot. It is a very complicated and hard to scale option. If you ever want to do something the non-magento way, you are looking at lots of time and developer cost. Also, as other's have pointed out, it seems like more people are leaving Magento than staying with it.
If you are looking to really customize would recommend Spree Commerce. It is a Ruby on Rails based cart. Spree allows you to customize your checkout and almost every other aspect of the cart. With version 2.0 you can use spree as just an API and have an interfacing app sit on top of it. Spree can be tested with automated test like rspec. You can also deploy it to a scalable solution like Heroku.
Hope that helps,