There are plenty of solutions for starting an eCommerce site with WordPress. However, the best solution for you will depend on what exactly you are selling.
If you are selling physical goods that will be shipped, it's hard to beat WooCommerce. If it is overkill for your needs, you can customize it and reduce it to only the features you want.
If you are looking for something much simpler, take a look at WP Simple Pay Pro. Easy way to integrate Stripe checkout with your site.
Another simple plugin I would recommend is Plasso, https://plasso.com/wordpress. Although Plasso is new to the WordPress game, their payment solutions have been around for a couple of years. They offer hosted pages for your products or services which look awesome.
If you are going to have subscription-based billing, I would recommend Zoho Subscriptions or Chargebee. They both offer hosted payment/checkout pages.
If you have any questions regarding these or other eCommerce solutions, or need help implementing them into your site, give me a call and I would be glad to help :)