I need the date that the New York Yankees went to Galveston, Tx to
play an exhibition baseball game.

Hello Ken,

According to the online archives of the New York Times, the Bronx
Bombers paid a visit to Galveston on March 14, 1914 to play (and
lose!) a 90-minute exhibition game before a crowd of 1,000 fans
against the Texas Leaguers of Galveston.

The NY Times archives can be searched here:

http://pqasb.pqarchiver.com/nytimes/advancedsearch.html

and a search on [ Yank* galveston exhibition ]

will pull up an article from 3/15/14:

YANKEES DEFEATED BY TEXAS LEAGUERS:
Lack of Pinch Hitting Very Apparent to Manager Chance in His Team. 
Mar 15, 1914.  pg. S1 

GALVESTON, Texas, March 14. -- The baseball patrons of this little
town turned out in large numbers to-day to see their home club mix in
a ball game with a big league team, and it was also in honor of Harry
Williams, who started his baseball career here and then advanced to
the big league....

----------

I can't copy the entire article here due to copyright restrictions,
but here are a few key excerpts:

...The Yankees were weak in the hitting end of the game once again,
which will mean a great deal of batting practice for the last week of
training.

...Ed Sweeney reported to the club before the Yankees left for
Galveston yesterday...He is a little fat...but will no doubt be ready
to play in the exhibition games the Yankees play on their way back
home...

----------

I couldn't resist including that "He's a little fat..." barb.

Hope this is what you need.  Any questions, just post a Request for
Clarification to let me know how I can assist you further.

pafalafa-ga

This information might interest you:

On April 9 1965, the New York Yankees played the Houston Astros in
exhibition baseball at the Astrodome, also known as the Harris County
Domed Stadium in Houston. Mickey Mantle hit the first homerun in the
Dome, but the Astros won.
http://tigerx.com/history/0409.htm