Central Liverpool is compact and best explored on foot. The walk from the Anglican Cathedral at one end of Hope Street to the Metropolitan Cathedral at the other takes 15 minutes; from the Pier Head to the Liverpool ONE shopping district is five minutes. Don't bother with a car in the centre.
Merseyrail is the suburban rail network, with frequent electric trains running underneath the city centre. It is the easiest way out to Crosby (for Antony Gormley's Another Place), Aigburth (for Sefton Park), or Birkenhead (for the Wirral). Buy contactless or with a paper ticket.
Buses run from Queen Square and Liverpool ONE bus stations to the suburbs and to football grounds; the 86 service is useful for Penny Lane and Mossley Hill. The Mersey Ferry crosses to Birkenhead and Seacombe from the Pier Head — slow, but the only way to see the city from the water. Taxis and Uber are plentiful in the centre.