What is the easiest way to reach yingkou from Shenyang Taoxian airport ?

You can take Shuttle Bus line 2 to Bus Station and then take an intercity bus to Yingkou. The shuttle bus takes about 50 minutes and CNY15, the latter one takes about 2h 30 minutes and CNY63.

Thanks Gloria. Any high speed train option to yingkou after landing at the airport?

Yes, there are bullet trains available from SY Railway Station and North Railway Station. The bullet train takes about 1h, CNY80.5/78.5 for a second class seat, CNY129.5/126.5 for a first one.
You can take Shuttle Bus Line 2 to the bus station, and then walk northeast for 10 minutes to the railway station.
To the north station, take Shuttle Bus Line 1there. It takes about 1h and CNY17.