Samuel, what version of Ionic are you using? Will you persist with this data in a database? Or is it all local?
In general, what you need is to have a screen for the listing (array) of the available products in which you have a search bar (for product filtering), in which each item of this listing (via ng-repeat or ngFor, depends on your version) will be a unique product, you can put its name and a "+" button (to add) to your list.
The click event of this button will execute a function that has as parameter the product item, and this parameter will be added at the last position (.push()) of the product array.
A very simple type of app that can help you are the "todo lists" (to-do list). You may end up basing yourself on one of them to make your.