Liverpool John Lennon Airport (LPL) is only about 10 miles to the city center. There are a few ways to get to the city center from just outside of the arrivals area; obviously, a taxi will always be the fastest (~20 minutes) and priciest (£12) option. There are also several buses that go to the city center (all take ~45 minutes):
The #500 Airport Express runs every 30 minutes (Children £1, Adults £2.60, Families £5)
The #80A runs every 15 minutes (£1.90)
The #82A runs every 30 minutes (£1.90)
The #86A runs every 15 minutes (£1.90)
Last but not least, if you hate security lines, LPL has a Fast Track service that allows you to skip the security line… if you like to cut it close, check out this service!
If you are having trouble finding flights to Liverpool, Manchester Airport is less than an hour away and could prove to be a great alternative.