Yes, you can get the new COVID-19 vaccine and this year's flu shot at the same time. The CDC says you can either get both shots in the same arm, at least an inch apart, or one shot in each arm. If you choose to get the shots at different times, there's no recommended waiting period — you can schedule one appointment shortly after the other.
Studies suggest that getting the shots together comes with a similar risk of side effects as getting a COVID-19 vaccine on its own; common side effects include headache, muscle ache and fatigue. The same research suggests that getting the shots together does not undermine the effectiveness of either vaccine.