@Jacquie0221 I echo @JaneCares advice to start with contacting your local area agency on aging and asking about transportation services. Some area agencies on aging have a list of senior transportation providers in the area, and you can find something there.
Some services take you to appointments and then come back when you call them to say your appointment is over and you're ready to go home. Other services offer "escort" where they will help you in to the appointment and wait with you. You can find out if there is a cost for the service (sometimes the fees are based on your income and ability to pay).
Some car services do offer more of a personalized service where they will wait for you while you shop, are in appointments etc. Taxi companies sometimes have this set up or black car companies. But be very careful and if possible use one of the services the area agency on aging has a relationship with because then there is more assurance the company and drivers have been vetted and less possibility of scammers or people who will take advantage. In this time of COVID-19 it is also all the more likely that they will be following safety protocols to keep the cars clean between passengers, wearing masks etc. (SO important!)
You can also use rideshare services like Uber or Lyft, and in some areas they partner with organizations to provide special services for seniors. But if you do that be sure they are following safety protocols for COVID-19. If you want to go that route, you might try using a service like GoGo Grandparent - a service in which you can call to get a rideshare instead of using a smartphone. They also offer communication with caregivers.
If you are 60 years of age or over, you can get either free or very low cost rides from your local area agency on aging. You can look that up at the Eldercare Locator, which is run by the federal government and helps you to find services in your town or county: https://eldercare.acl.gov/Public/Index.aspx.
If you want a personal driver, that's another thing...
Would the service by your local Senior Center do the trick?