Transportation

From Nagoya Chubu airport to Nagoya: 

The fastest way is the mu-Sky limited express which is part of the Meitetsu line. It takes 28 minutes from Nagoya airport to Nagoya station, and costs 1230Yen. 

You can consult this webpage, which has detailed information and pictures. 

You can then change train in Kanayama or Nagoya Station for instance, depending on your final destination.

Taking the subway in Nagoya:  

For information about how to use the subway in Nagoya, please consult this website.

From Osaka Kansai international airport (KIX) to Nagoya station: 

The fastest way is to take the Haruka Limited Express train from KIX to Shin-Osaka, and then to change for the Tokaido-Sanyo Shinkansen, towards Tokyo. For the second train, the fastest shinkansen are the Nozomi trains, which go every 3 to 10 minutes. In that case, Nagoya will be the second stop (after Kyoto).

You don't need to book a ticket in advance, as the trains are rarely fully booked in advance. For both trains, you need to buy both the regular ticket and the Limited-Express supplement. For the shinkansen, there are two kinds of tickets, reserved seats and non-reserved. The price for the Haruka train is 2330Yen, and the price for the reserved Nozomi is 6560Yen.

For the Haruka train, you can consult this website.