I apologize, evidently Korea slipped my mind. They most certainly do have AH-64's there.
As for what to fly? Personal choice is all you have to go on. Once you progress through IERW, just after instruments, you should get your AQC. They may give you a couple of choices, but then again, they may give you only one. How you do in IERW through Primary and Instruments will help you in getting the aircraft of choice.
I personally would fly 60's, because that is what I have around 1,000 hours in, however, each aircraft has its highlights and nuances. What you have to think about is the outside. Helicopter pilots are not in HUGE demand. The people that get these slots keep them until they physically cannot fly any longer.
The most common outside aviation jobs are flying to offshore oil rigs in the gulf, EMS transport, corporate transport, and tour helicopters. If you would like law enforcement, it is rarely available, but you still have to do road time last I checked. DEA and ATF use them as well as Customs and Border Patrol.
My advice, have a fall-back degree. Although aviation service is rewarding, it is also limited in civilian use. One other option - when you become a senior CW2, put in for the fixed-wing transition. You can transition to C-12's (King Air's), RC-7's (Dehavaland Dash-7), UC-35's (Cessna Citations), or if you are lucky, the coveted Gulfstream fleet in which I currently am an engineer on.
As always, the choice is yours; it is your career. Make the most of it, but do it with the knowledge you need. Never assume anyone will tell you what is best or help you attain it. Aviation is not like other Army branches......
UH-60's as you said, would afford you the most duty possibilities, but OH-58's will soon be replaced by the new ARH, which from what I hear is a beasty little bird (its a super-beefed-up Kiowa).
Whatever you choose, have fun and stay Army! (thats for ArmyReenlistment).
OIF 1 VET
22 Dec 02 - 18 Aug 03
A/B Co. 2nd BN/3rd AVN Regt.
3rd Infantry Division (MECH)
AEF 8
OIF/OEF
15 MAR 06 - 7 JUL 06
USAPAT-PFD