Hi, here's another thumbs up for Manchester Airport. It is very easy to use. From experience, immigration lines at 6am are usually very short, and you'll be through very quickly.
If you all need a break before heading out to Liverpool, can I recommend a quick breakfast stop at Cafe Ritazza at the train station. It is always much quieter than the cafes in the terminal building, and there's space to let the kids run around.
There is a covered moving walkway from the terminal to the train station, which like someone else said, is very easy to navigate - only two tracks. See
www.manchester airport.co.uk for maps and more info.
There is a direct train from the airport to Liverpool every hour and the journey takes 1 hour. Check
www.nationalrail.co.uk for times. It's £9.40 one way or £12.20 round trip. The kids will be free. There is no 1st class seating on this train, as it is a local train.
If you don't feel like waiting for a direct train, take any train from the airport to Manchester Piccadilly (the main station in central Manchester) - there are trains every 10 mins. At Piccadilly, change for a Liverpool train - they run every 15 mins - same price ticket. If you go this way, you can travel 1st class (more room) for £11.50 (one way) - so not a big difference, but you might appreciate the space with the kids and the luggage etc .....
You'll need to decide at the airport station whether you want to go first or second class ...
OK, have fun!