Shared is cheapest, and thus best suited for very small businesses. As you grow and get more visitors and traffic, eventually you'll have to move up to VPS or even dedicated to have the needed resources.
For a basic store using a Third Party payment system (e.g., PayPal), an SSL certificate (requiring a dedicated IP address) is usually all that's needed (for customer passwords, shipping information, etc.). Note that if you want to directly accept credit cards (i.e., handle the card number on your site) with a payment gateway and merchant account, you will need to pass a PCI-DSS audit of your security systems. You'd have to ask LP if their shared servers have passed such audits, and if so, what was the cost to the storeowner. If it can't pass the audit, you'd have to look into starting with a VPS or even a dedicated. Frankly, between the audit fees and the fixed overhead costs of a merchant account, it's usually cheaper to go Third Party for small stores. At some point in sales volume, their higher per-transaction fees begin to exceed the audit cost + fixed fees + low per-transaction fees of a merchant account, and you should consider switching over (which may involve upgrading your hosting plan).
A special note for brick and mortar stores branching out into e-commerce: Do not accept credit card numbers via the e-store and enter them into your store POS system. You will be violating your merchant account system agreement and may be breaking the law. There are some Third Party systems that for a small fee (or even free) will accept and hold the credit card number (with permission of your bank); they notify you when you have a customer charge and you sign on to their system to see the card number and manually enter it into your POS system. Your store never sees the credit card number, and so doesn't have to be PCI-DSS compliant.