Should I use a standard mileage rate or actual expense method? (old and new car)

It depends.  I would suggest using the standard mileage rate (SMR) for the old car for sure.  The rate for 2023 is 65.5 cents per mile.

Enter two separate vehicles. First the old car and then the new car.

• Review IRS Publication 463 (Chapter 4)

For the new car, it's important to note that if you don't use the SMR the first year you place a car in service, then you can never use it on that car again. 

There is a limit on the total section 179 deduction, special depreciation allowance, and depreciation deduction for cars, trucks, and vans that may reduce or eliminate any benefit from claiming the section 179 deduction. See Depreciation Limits (page 29).  Section 179 is an option if you have high income, but if the income is lower in 2023 you may want to either depreciate the car over 5 years or take the SMR to give yourself options and it's a lot less recordkeeping.  Simply track your business miles each year with an app or a computer file such as excel.

The old car may have some tax consequences if you sold it or traded it for the new car.  If you did not sell or trade the old car there is nothing to do until you do.

• When you use expenses for your car to reduce your income then a sale or trade becomes a tax event to determine gain or loss on the exchange strictly for the business portion.  Total business miles all years divided by total miles on the business vehicle all years give you the business use percentage to use on the sales price or trade-in value. The following information will show you how to handle this in your tax return.  

OLD VEHICLE Actions if sold or traded:

The way to report the sale (trade is not really recognized by the IRS any longer for equipment or vehicles) is as follows. You have all the records so it should provide you the detail to move forward.

1. All business miles for all years and then total miles for all years - divide business miles by total miles to arrive at your overall business use percentage for the life of the vehicle. You will use this percentage times the selling price (or trade-in value) to arrive at the business selling price. Use the same percentage to arrive at your business portion of the cost before depreciation. 
2. You used the standard mileage rate and a portion of that rate represents depreciation. See the chart below to arrive at the depreciation portion for the business miles only.
3. When you go through the vehicle information in your rental be sure to select 'Sold, disposed of, etc....' then do not indicate it was sold.  You must say 'Yes' it was converted to personal use.  This will eliminate any sales information in the vehicle it self.
4. Once you have completed the information in that section you will follow the steps below to enter your sale:
   1. Go to Less Common Business Situations
   2. Scroll to Sale of  Business Property
   3. On the next screen select Any Other Property Sale
   4. Use the information from step one and the depreciation from step 2 to complete your sale
5. If the personal portion of your vehicle is a loss there is nothing to report for that portion of the sale/trade.