From La Guardia Airport to connecting train to the Bronx?

Unfortunately, the James J. Peters VA Medical Center is not right by a subway station, so if you're carrying a lot of luggage, I'd just take the taxi straight there.

If not, I'd take the taxi (or the M60 bus) to 125th Street/Lexington Avenue and then take the Bronx-bound (4) train to Kingsbridge Road. From there, you have to walk about 1/3 of a mile (8 short blocks) westward to reach the hospital. (Or you can take the Bx9, but it won't save much time)

You can buy a $4.50 MetroCard for the trip at the Hudson Newsstand in LaGuardia Airport. If you're just going to take the subway, a SingleRide Ticket costs $2.50.

I hope this information was very helpful.

Your best bet from LaGuardia Airport would be take the M60 bus into Manhattan, get off that at 125th street @ Lexington Ave in Harlem, and get a Bronx-bound #4 subway to Kingsbridge Road, then walk a few blocks west to the VA Med Center. Or get the Bx9 bus from Kingsbridge Road to the VA Med Center.

The directions Nick gave works too.

Take the cab to the Junction Blvd stop on the 7 train, Take it towards Manhattan and Times Square. Get off at Grand Central and transfer to the uptown 4 train, it'll say towards uptown and the Bronx. Take the 4 train to Kingsbridge Road, you'll have to walk west on Kingsbridge for about a third of a mile.

It'll take probably an hour and a half to do all this.