I did this exact thing a couple years back when I found a bank offering way better savings rates. The process is pretty straightforward, but timing matters. You'll want to open the new account first and get it fully set up before you touch anything at your old bank. Once you have your routing number and account number from the new place, you can update your direct deposit through your employer's payroll system yourself - no need to loop in HR or anything. Most payroll portals let you do this online, and if yours doesn't, a quick email to whoever handles payroll works fine.
The timeline is usually smooth. I updated mine on a Wednesday afternoon, and my next paycheck hit the new account right on schedule two weeks later. Sometimes there's a delay if you catch a payroll cycle that's already processed, so ideally update it a few days before the payroll cutoff if you can. Don't close your old account immediately - keep it open for at least a month after the first deposit lands in the new place, just in case something goes sideways and you need to troubleshoot. After I confirmed everything was working, I kept that old account active for another two weeks before closing it out.
One thing I'm glad I did was test it with a smaller paycheck first if possible, or at least verify the deposit went through before the big paycheck arrived. That gave me peace of mind. The whole process was way less stressful than I thought it'd be.