Easy - there's trains direct from Manchester Airport to Lancaster, every hour on the hour, taking 1h13. There's also an indirect route (changing at Preston) at 29 past, taking 1h25.
Walk-up fare is £17.10 after 9.30am, pre-booked fare is £6.50 in standard, £11.50 in first. Trains are operated by
TransPennine Express - nice, new trains, but can get busy.
You can pre-book tickets from abroad using
East Coast's booking engine, and pick them up at Manchester Airport station.