Shopify allows you to create up to 3 product variants, such as Size, Color, and Material for any given product. If you require additional options or custom input, you'll want to check out Shopify Line Item Properties.
Instead of adding your new custom fields to product.liquid, you'll want to add them to product-form.liquid above the 'add to cart' button.
There are some limitations with line item properties. Unlike product variants, you cannot track inventory or change the product's price with your additional custom fields.
For example, if your product supports an optional monogram that would have an additional cost, then it's recommended to create a variant called 'Monogram' with options 'None' and 'Full Name'.
The 'Full Name' option could change the price of the product. You can then use a custom field to support the optional monogram as customer input.