Just trying to understand your question better. You're looking at a way to take your current biz online , would that be right?
So what you're looking for is an web presence where your users can visit, have a look at various categories or options of t-shirts for example (men, women, kids, or maybe theme -based designs ), customize a t-shirt by choosing a design, then their t-shirt size, and then proceed to payment and complete that as well. You need to be able to handle your invoicing, payments and keep track of your shipments.
If the above is what you need (or somewhere close to what you need :-) ) , then there are different options based on the time you have and need. There are many customizable e-comm solutions (think of them like templates that support various functionalities like a shopping cart, payment, and what not).
Shopify is one such example. Magento is another. Check out Shopify and see if it suits your needs.
Let me know if you have further questions! Cheers and good luck!