My name is Lizzy and I’m from the support team at Shopify.
You can absolutely set something like this up! This page from our help docs lists all of the details you can add to your products. Weight is listed under Shipping. If you have a product without variants you will be able to input the weight here.
If your products have variants you will need to input the weight of each one. You can do this using the bulk editor either on the main products list, or within an individual product page. This page here has a video showing how to use the Bulk Editor in your admin.
Once you've got your product weights inputted, you can choose how you'd like to set up your shipping rates. If you want to set up manual rates we have a great page in the Shopify Help Center which you can read here. Manual rates give you the option to set up how much your customer is charged based on the weight of all products in their cart plus the weight of your saved parcel.
There are many different ways you can set up your shipping rates that may help to drive sales. Have you had a chance to think about your shipping and pricing structure? Have you considered offering free shipping and including the shipping cost in your products?
If you're located in Canada or the United States you could use Shopify Shipping to set up calculated shipping rates. These would automatically use the weight of your products and package to charge the customer a calculated shipping rate based on their address. This page here has some more information about how that works.
If you are located outside of those countries, or are hoping to use a carrier that isn't included in Shopify Shipping you also have the option to set up Third Party Calculated Rates. This is a feature which gives outside rates access to your shipping settings at checkout. It is included in the Advanced plan, or available for a $20/month add on to our other plans. It may also be available for free on an annual plan. This page here explains how it works. If you're interested in setting this up, let me know and I can start the process for you!
Let me know if you have any questions at all about what I've said!