Private car to taxi

If you want to do it more cheaply, there are train and bus (coach) options.
https://www.manchesterairport.co.uk/getting-to-and-from/liverpool

Otherwise, this taxi site may help you.
https://airportstaxitransfers.com/transportation/taxibookings/Transfer-from-Manchester-Airport-to-Liverpool-City-Centre

any reason for not taking the train? Trains leave right from Manchester Airport and go right into Liverpool. Perhaps a taxi from Liverpool Lime Street to your destination? Just over an hour, 6 stops, no changes.

For the distance you want to drive I suggest getting a pre-arrranged private car (minicab) rather than a random cab.

71 yr old woman with large suitcase. Traveling alone to hotel at Albert Docks. I think a car wbe better me than train. I have done trains in Europe before and sometimes it is a drama!

Trains are a pain--- take it from 2 young individuals trying to get 3 pieces of luggage up a large gap..no fun!

Any recommendations for car companies?