The training arena is in the online section of the game. Start the game, log in, then look at the screen and it will show all the arenas available for your use. The Training Arena (TA) usually has a small group of pilots in it, some who are also learning, and others who are there to try and help out. Feel free to ask questions, or ask for help, when in there as there are usually people who may have the same questions, or people who can answer them. Also, look for members who are in the Training Corps (I'm a member), they often transmit on the radio in a blue colour. They are specifically there to help people out.
There is no store version of Aces High, you have the full version from the download, and if there are updates it will automatically prompt you when you log in to download the latest version. If you version is fairly seriously out of date it may take a little while, but if you are simply missing the latest patches, it shouldn't take long at all. Maps will also download as required when you enter an arena for which you don't have a map, or for which a new map is available. Simple as can be. If your download connection is slow, I think you can order an version on CD from HTC, but typically this shouldn't be required (I don't know anyone who has ordered one)
-Soda
AH Trainer
The Assassins.