In general you should wait two weeks between getting Covid vaccine and the flu vaccine. Because of Covid and all the social distancing measures that have been in place, cases of influenza this year are way down this season. But it is still recommended to get it. Check with your healthcare provider. There’s also a delay in getting the Covid vaccine if you’ve recently had the Covid infection.
He says his arm's a little tender, I can't even feel mine. But a funny thing happened when I got my shot .... my phone was on my lap, and just as the needle went in, I felt something like an electric shock go through me, not once but twice! DH had said he didn't even feel the needle so I thought that was really odd - until I realized I was getting a text and it was my phone vibrating! 
