To have an online store, the easiest way is to find some open source ecommerce script. For example oscommerce or zen-cart shopping cart. Get these script setup on your web hosting.
To accept payment online, its either customer pay through your website, or route customer to payment gateway website and make payment. Selecting which one, will depend on your needs and budget.
Having third party merchant is easiest and lowest cost. You could sign up with 2checkout payment processing or paypal or verisign payment processing. And route customer to the link provided by them. Customer will make payment at those provider webpage, and once its done, customer will return back to your website.
Be sure to pick only those famous third party merchant provider, don’t trust those that you found online and offering incredible low cost and no fees solution. Who knows, you will never get your money at the ends.
Either you can find a web hosting provider that offers merchant account as well, and they will handle all the integration work for your ecommerce store.