Good advice above.
I like the M1903...it is a fine battle rifle and an great military target rifle. There are lots of variations...03, A1, Mk I. The sights are highly adjustable but are of an older generation. Springfields, Rock Island, Smith Corona, Remingtons. I have a Rock Island that is only a couple of thousand outside of the ones with the heat treat problems.

The 03A3 is a better rifle from a shooter's view than the 03 in my opinion. While bottom metal is stamped, the sights are better...aperature and a longer sight radius. Mine is a Remington.

Don't forget about other US Military arms. See the Krag above and the 1917. The Krag is a cool design and has a butter smooth action. The 1917 is a classic Enfield action...pot belly and all. Rumor is that Sgt York actually used a 1917 rather than a 1903 in his heroic event.
Also don't forget about other military bolt actions. Everyone loves German K98 Mausers and I have a few...but they aren't my favorites. I really like Swedish Mausers, Finn M39s. Don't care for Mosins that much (even though the M39 is based on one) K31s are fun too. The Arisaka is actually a very nice rifle...underappreciated in my opinion.

British, French, Swedish and Swiss WWII rifles
Ishapore Enfield 2A, 7.62x51mm
Savage Enfield No. 4, Mk I*, .303 British
Fazerkly Enfield No. 4, Mk 1/2, .303 British
MAS 36, 7.5x54mm
Berthier Mle 1892/27 (32 ?) (Mosqueton d'Artillerie Mle. 1927), 8x50R mm
Swedish M1896 Mauser 1900 Oberndorf, 6.5x55mm
Swedish M1896 Mauser 1903 Gustav, 6.5x55mm
Swiss K31, 7.5x55mm
Swiss K31, 7.5x55mm

Yugo, German, Japanese, Russian, Finnish military bolt actions
Yugoslavian M48 Mauser, 7.92x57mm
Mauser K98k, 7.92x57mm
Mauser K98k, 7.92x57mm
Type 99 Arisaka, 7.7x58mm
Mosin-Nagant M38, 7.62x54R
Mosin-Nagant M44, 7.62x54R
Mosin-Nagant 91/30, 7.62x54R
Finnish M39 Sako, 7.62x54R
Finnish M39 B-barrel, 7.62x54R