I would like to set the shipping date for my orders to the next day, as many times I'm packing orders late at night after post offces close. When I used to fulful orders via Paypal, they would give me the option to choose one to two days out for the ship date. Can this be done in Shopify?
There is no way in Shopify natively to "set" a shipping date. You may be able to achieve this with a shipping app, but you'd need to have Carrier Calculated Shipping as a part of your plan to use most apps that have control over shipping rates like that.
I would suggest changing language on your shipping rates, adding text somewhere on the site, as well adding shipping information in your "refund policy" and "terms of service".
In response to what Alicia had stated, if you're looking for a solution to this, and potentially for your rate calculations, Intuitive Shipping is an advanced calculation app that gives the option to add processing time to the carrier delivery times. There are also options to customize this, too, if different products or groups need different processing times. If you have any questions about it, please let me know!