A trained-for-war animal's stat block remains mostly unchanged
A heavy horse, light horse, or pony (Monster Manual 273, 273, and 277, respectively) that's trained for the purpose combat riding isn't transformed into a heavy warhorse, light warhorse, or war pony (273–4, 274, and 277–8, respectively). Instead, a heavy horse, light horse, or pony that's trained for the purpose combat riding gains the tricks attack, come, defend, down, guard, and heel; a rider atop such a creature can use the Ride skill use fight with warhorse; and—in the only real change to such a creature's stat blocks—the creature can use its hooves as primary attacks.
The Player's Handbook mentions nothing in the description of the skill Handle Animal (74–5) about how training an animal for the combat riding purpose fundamentally changes anything about the creature except that the creature will now perform the tricks it knows for its handler more readily. Similarly, the description of the Ride skill says a rider can make a Ride skill check to fight with warhorse if the mount is "war-trained" (80). Finally, only the Monster Manual mentions—and, then, only in passing—that
A horse not trained for war does not normally use its hooves to attack. Its hoof attack is treated as a secondary attack and adds only half the horse’s Strength bonus to damage. (273)
(A similar note is provided for the pony (277).) This strongly implies that a trained-for-war heavy horse, light horse, or pony uses its hooves to attack normally as primary attacks just like the respective heavy warhorse, light warhorse, and war pony.
When an animal is trained for the purpose combat riding, it knows some tricks and a rider can fight atop it more readily. Plus, if it's a heavy horse, light horse, or pony, it can fight with its hooves normally. Otherwise, the animal is unchanged.
(In the above, I assume—based on my research—that trained for war and war-trained are synonyms for the special purpose combat riding. (Your own research might yield different results, and, if it does, I'd love to hear them. I find the mount rules utterly opaque without if I don't make that assumption.) Also, I urge reading all four the articles "All about Mounts"; while Rules of the Game columns can be sometimes inaccurate, these articles are also one of the handful of places in the game's corpus that discusses mounts at all, much less in detail. Sometimes these "All about Mounts" articles in particular shed light on otherwise impossible-to-deal-with issues like this and this.)
"How can a light horse become a light warhorse?"
Technically, using mundane methods, it can't. A light horse, for example, is a different creature from a light warhorse, with not just a different price (75 gp versus 150 gp according to PH 129) but different ability scores (Str 14 versus 16 and Con 15 versus 17) (and—if it makes a difference—a separate Monster Manual entry).
For example, spending 6 weeks and succeeding on a Handle Animal skill check (DC 20) to train a light horse for the purpose combat riding will not grant that light horse Str +2 and Con +2.
(Want an animal with different ability scores? Ask the DM, when shopping for an animal, if there's an animal available that uses instead the nonelite array or even the elite array… then expect to pay a lot more for that animal! Alternatively, convince the DM that an animal already in your possession has become more than a possession and should be considered a cohort (DMG 104–5) so that the animal can gain XP alongside its master and advance in Hit Dice or level. (N.b. a light horse, for example, has the entry Advancement: —, so it and others like it—except in rare cases—never advance.) Finally, the 4th-level Drd spell awaken [trans] (PH 202) opens the door wide to a host of possibilities.)
…And about that warbeast template
This DM urges against using as written the template warbeast (MM2 219). In addition to the template warbeast requiring the creature to which the template be applied be reared by a handler for 1 year then trained by a handler for 2 months more, instead just buying a warbeast creature is one of the game's best—and most broken—deals. Even going by the Dungeon Master's Guide's "general rule" that a PC can't spend more than half his starting wealth on one thing (42), one 2nd-level PC can still afford a warbeast animal or vermin that possesses up to 12 Hit Dice, like a warbeast elephant (MM 272–3) or a warbeast Huge monstrous scorpion (287–8) (both typically CR 8). Consider carefully what kind of campaign you want to run before introducing into the campaign the template warbeast: a campaign that makes such high-powered creatures available so inexpensively looks and plays a lot differently from one that doesn't.