Training the animal faster using only the Player's Handbook and the rules-as-written

This is probably impossible. The Handle Animal skill says

For [Handle Animal skill] tasks with specific time frames noted above [like the 1 week it takes to train an animal in 1 trick], you must spend half this time [i.e. half a week, or 3 days rounded down, for training 1 trick] (at the rate of 3 hours per day per animal being handled) working toward completion of the task before you attempt the Handle Animal check. If [on the third day] the [Handle Animal skill] check [attempt] fails [at that point], your attempt to teach, rear, or train the animal fails and you need not complete the teaching, rearing, or training time [and you can start over]. If the [Handle Animal skll] check [attempt] succeeds [on the third day], you must invest the remainder of the time to complete the teaching, rearing, or training. If the time [spent, like, training] is interrupted or the task is not followed through to completion, the attempt to teach, rear, or train the animal automatically fails. (Player's Handbook 75)

(Yes, that's a lot of brackets.) This DM reads this description as there just not being a way to, for example, increase daily training to reduce weekly training. This DM suspects that were that an option, it'd be explicit in the skill's description.

For instance, a handler usually takes 1 week spending 3 hours per day to train 1 animal in 1 trick. A handler doesn't seem to have the option to, instead, take 1 day yet spend 21 hours during that day to teach 1 animal 1 trick.

If animal training isn't a big deal in the campaign, the DM may be amenable to a more liberal reading, but if animal training is a big deal, steel yourself for a No.

Other sources allow Handle Animal to be used to train a creature more quickly

Although epic Handle Animal skill uses aren't listed with the typical uses of the skill Handle Animal, neither the SRD nor the Epic Level Handbook prevents nonepic creatures from trying to employ epic skill uses. Further, the "Epic Level Handbook Frequently Asked Questions" (Apr. 2003) includes this exchange:

Can a character who has not reached epic levels achieve the results listed for an epic level skill DC? For example, suppose my 10th-level wizard could achieve a Spellcraft result of 50 through some combination of ability bonuses and skill bonuses; could the character identify potions quickly?
Yes. Anyone who can manage a sufficiently high skill check result can get an “epic” result. (2)

However, despite their presence in the SRD, epic skill uses for the skill Handle Animal were insufficiently modified for the 3.5 revision to the skill Handle Animal. This means, for example, when the description of the epic Handle Animal skill says that "[n]ormally, teaching or training a creature requires two months of time," that's according to the dnd-3e rules rather than dnd-3.5e rules, the dnd-3.5e rules having shortened this normal time to a more reasonable—yet still burdensome—1 week per trick. (Updating was probably complicated by dnd-3e tricks appearing not in the Player's Handbook but in the Dungeon Master's Guide (46).)

That said, the DM should make "minor adjustments" to the epic uses of the skill Handle Animal according to the Dungeon Master's Guide (2003) on Why a Revision? (4). This DM suggests the chart be adjusted as follows:

| Reduce Training Time Per |      |
| Trick from 1 Week to…    |  DC  |
|--------------------------|------|
| 1 day                    |  +25 |
|--------------------------|------|
| 1 hour                   |  +50 |
|--------------------------|------|
| 1 minute                 |  +75 |
|--------------------------|------|
| 1 full-round action      | +100 |
|--------------------------|------|

(This DM includes as a training time 1 full-round action because A) that seems to him a logical progression given the chart, and B) that's awesome; another DM may hold a different opinion.) Thus, for example, using the above chart, teaching an animal the trick defend (PH 75) in 1 day requires a Handle Animal skill check (DC 45), and teaching a spider eater (Monster Manual 234) the trick perform (PH 75) in 1 min. requires a Handle Animal skill check (DC 95). Such changes still make it challenging for even an exceptional handler to teach a trick quickly, but the results become possible with effort, ingenuity, and, of course, money.

Employ hirelings to help train the creature

The easiest way for a handler to increase his Handle Animal skill check result is to have others aid another in making the Handle Animal skill check (PH 65-6). For 3 sp per day a trained hireling can be employed (PH 129, 132).¹ Every 3 gp spent means for 1 day a +10 bonus on the handler's Handle Animal skill checks even if half the hirelings fail their checks to aid the handler.^2,3 However, a DM may, rule that "only a limited number of characters can help at once" with the handler's Handle Animal check (PH 66), so, for example, the DM may rule that it's simply not an option to pay 36 gp to get 120 trained hirelings to help you teach your dog 5 tricks in 30 seconds.

Use magic to help train the creature

In addition to the variety of spells that can assist generally in making skill checks (e.g. the 2nd-level Clr spell divine insight [div] (Spell Compendium 70), the 2nd-level Clr spell guidance of the avatar [div] (Spellbook Web column "Guidance of the Avatar"), the 4th-level royal herald spell sudden mastery [trans] (Dragon Annual #5 87)), two spells specifically help with animal tricks:

• The 0-level Drd spell animal trick [trans] (Masters of the Wild 82) takes a standard action to cast and causes a caster's animal companion that's within close range, at the next available opportunity, to perform a trick it doesn't know. Note: The DM may make "minor adjustments" to the spell as it is untouched by the 3.5 revision.

  If using the Rules Compendium's rule that allows a spell generated by a magic item to be generated as a swift or immediate action if the spell's casting time is likewise (see RC 84), a wand of quickened animal trick (0-level spell at caster level 1 modified by the feat Quicken Spell (PH 98) to a 4th-level spell requiring caster level 7) (21,000 gp; 0 lb.), while terribly expensive, may be exactly what you're looking for.

• The 2nd-level Drd spell train animal [ench] (SpC 221) takes 10 min. to cast and for 1 hour/level grants a touched animal a number of additional tricks equal to half the caster's level (maximum 5), presumably even beyond the normal limits of 3 tricks for an Intelligence 1 animal or 6 tricks for an Intelligence 2 animal. However, a typical creature must still take a move action to make a Handle Animal skill check (DC 10) to handle the animal; failure means that the animal doesn't perform the trick.

Further, the 1st-level Drd spell charm animal [ench] (PH 208), the 1st-level Drd spell speak with animals [div] (PH 281), the 3rd-level Drd spell dominate animal [ench] (PH 224), and similar spells should facilitate, to one degree or another, bossing around an animal so that it does what the caster wants.

Finally, the neck slot item collar of obedience (Complete Adventurer 132) (1,500 gp; 0 lbs.) reduces by 5 "[t]he DC of any Handle Animal check [made] to handle, push, teach, train, or rear the wearer" if the wearer's an animal or magical beast, therefore substantially increasing the odds that one's hirelings, if trained but largely inadequate, will succeed on their Handle Animal skill checks to aid another. (Undoubtedly other magic items that grant bonuses to Handle Animal skill checks exist in Dragon magazine and elsewhere, but a list of all such items is beyond this question's scope.)

"Can a really smart animal learn faster?"

Unlike Pathfinder, the game doesn't really accommodate animals possessing Intelligence scores of greater than 2. One of the traits of the type animal is that such a creature possesses an "Intelligence score of 1 or 2[, and ] no creature with an Intelligence score of 3 or higher can be an animal" (MM 306). What this means exactly when, for example, a master bestows upon his otherwise typical cat a headband of intellect +2 (DMG 258) (4,000 gp; 0 lbs.)—besides likely improving the cat's already healthy self-esteem—is a mystery best left to the DM, but the Player's Handbook on the skill Handle Animal says, "You can use this skill on a creature with an Intelligence score of 1 or 2 that is not an animal, but the DC of any such check increases by 5" (PH 75), so, in all likelihood, a high Intelligence score precludes a master from using the skill Handle Animal on the creature in any way, the smarter creature, instead, doing whatever it wants.

However, the Monster Manual also says that "[a]ny creature with an Intelligence score of 3 or higher understands at least one language (Common, unless noted otherwise)" (7). This DM tends to view this rule as applying only to creatures with unmodified Intelligence scores of at least 3. Another DM, however, may rule that, for example, a cat wearing a headband automatically understands Common. Either way, though, whether such a cat obeys commands is likely up to the cat.

¹ This DM assumes trained hirelings are necessary because the Handle Animal skill can only be used untrained on domesticated animals, this DM boldly assuming that, for example, a druid's newly acquired T-rex animal companion isn't domesticated.
² A commoner may've opted for the skills Listen and Spot instead of, for example, Handle Animal and Profession (sailor), but, as a commoner lacking the skill Handle Animal is in even more danger from a house cat than other commoners, this DM views at least 1 rank in the Handle Animal skill as necessary for any commoner's survival, making such trained hirelings easily found in this DM's campaigns. Ask the DM.
³ A creature "can’t take 10 on a skill check to aid another" (PH 66); odds can be increased by equipping each hireling with a masterwork tool (PH 128, 130-1) (50 gp; 1 lb.) for the skill Handle Animal, but keep an eye on those tools as crafty hirelings may abscond with them: it takes over 5 months working 7 days a week and not eating for a hireling to afford his own masterwork tool! (The DM may, instead, require an animal training kit (Complete Adventurer 122, 123) (75 gp; 15 lbs.); a generous DM allows stacking both circumstance bonuses.)