Many airlines allow each person to have two items with them.
A Personal Item.
A Carry-on bag.
It is worth being very aware of what the specific airliner policy is regarding these and how the policy may treat them differently.
The Personal Item is allowed to be substantially larger then one might expect. This could simply be a purse that does not have much room. Or this might be a larger item such as a backpack (potentially with a purse inside along with other things). I find a full size backpack with multiple compartments works quite well for the personal item.
United Airlines policy for a Personal item is 9" x 10" x 17" and it must be able to slide on the floor to fit completely under the seat in front of you.
United Airlines policy for a Carry-on Bag is 9" x 14" x 22" (45 max linear inches combined) and it must be able to fit into the overhead bin.
Delta policy is similar although not exactly the same as United. Note that Delta policy states that flights of 50 seats or less are only allowed the single Personal Item and all Carry-on bags will be gate checked.
It an airplane is unexpectedly tight on space, the airline may request the Carry-on Bags be gate checked however it is less likely for the airline to also request the Personal Item to be gate checked.
Although I am not a frequent traveler, when I have flown (myself and one other) we put all expected supply needs for 2 days along with carbs, food, drink, etc… into two Personal Items with each of us carrying one. Other supplies expected to be needed for the duration of the trip (past 2 days) go into the two (one each) Carry-on Bags. We do not pack any supplies into the checked luggage but rather put clothing and such into checked luggage.