What should I use to implement a shopping cart process?
Here are a few typical solutions for implementing an online store:

Drupal Commerce, or its related Commerce Kickstart.
Ubercart.

However, as per your &quot;Simple shopping cart experience&quot;, it looks like none of these is what you're looking for. So here are some other modules you may want to look at (excerpts are from their project pages):

Basic Cart:

... a very simple shopping cart / checkout process for Drupal, that just sends 2 emails after each order. Along with the shopping cart, this module also provides a block where you can view your cart's content. It's ideal for small websites with only a few products or other content types being sold, for example touristic offers.


Rules:

Allows site administrators to define conditionally executed actions based on occurring events (known as reactive or ECA rules).


Flag:

Flag is a flexible flagging system that is completely customizable by the administrator.



The 'Basic Cart' implements the cart, while with the 'Rules' and 'Flag' modules you can implement appropriate business logic (which does not require any PHP coding skills).
If you're not familiar (enough/yet) with Rules, then checkout the 32 (!!!) great, and free, video tutorials Learn the Rules framework, organised in 7 chapters.
Possibly also the similar set of 8 video tutorials about the Flag module, often considered as a natural complement to the Rules module.