Yes the USPS mail does travel on weekends.
USPS operates the sorting facilities and over the road USPS trucks everyday of the week and the sorting facilities are operating and open 24 hours per day and 7 days per week including most holidays.
The USPS keeps mail traveling so that the mail can arrive on time.
Saturday is considered a business day for USPS.
Mail is delivered on Saturday and post offices are also open on Saturday.
Some Post Offices are open on Saturday for half the day while some larger post offices are open normal business hours as well.
Sunday is not considered a business day for USPS since the post offices are closed although some mail such as on rural routes or larger cities may get delivered on Sunday.
The USPS sorting facilities are open on Saturday and Sunday and most holidays as well and mail is being sorted 24 hours per day and 7 days per week.
Mail also travels on the weekends and even Saturday and holidays to get to you.