It's not an easy journey, however you do it.
At certain times, you can take a train from Seaforth & Litherland to Moorfields, transfer to a service 10A bus and then walk the final part of the way.
That's the simplest route. At other times you can take a train to Bootle Oriel Road, change to an 81 bus, change again to service 10A and then walk.
If you don't want to take the train, you'll typically have to take 3 buses (144, 81A and 10A) and then walk.
You'll see that all of those ways of getting to the Safari Park end with a walk. This is because the nearest bus stop seems to be about a mile away (at Brookbridge Garage, on Liverpool Road). The estimated walking time is 19 minutes. The total journey time (including the walk) varies but it's typically around 1 hour 20 minutes.
All of this information is derived from the journey planner on the Traveline North West site (which searches all bus and rail timetables for the region):
http://nw.aimwebsites.co.uk/JourneyPlanner.jsp ?vm_Reset_str=true
Chris