In the Arms and Equipment Guide a young, untrained wyvern costs 3,000 GP. It costs another 3,000 GP to, instead, buy one that's trained. I want to train it myself. How do I do this? How much freedom will I have to determine the wyvern's feats? How loyal will this wyvern be?
In dnd-3.5e riders needn't bother training most intelligent creatures that can serve as mounts.1 Instead, a rider makes Diplomacy skill checks to negotiate with the intelligent mount about what the mount will do while the rider's aboard. An intelligent mount's loyalty is gained through role-playing as one would gain the loyalty of any NPC. And, as an NPC, an intelligent mount has whatever feats the DM gives it.
Training a wyvern
Mounts with Intelligence scores of 5 or higher [like, conveniently, the wyvern (Monster Manual 259)] are more like NPCs than they are like traditional mounts. As a result, characters must use Diplomacy checks to negotiate what the mount will and will not do (see Influencing NPC Attitudes, page 72 of the Player’s Handbook). Once the character and mount reach an agreement, they still must train together. Training time is as given under the Handle Animal skill (page 74 of the Player’s Handbook). Intelligent mounts may insist on special care, such as a hireling devoted to that specific mount, special food, or even a share of the party treasure. (205)
Following the rules for the skill Handle Animal, a trainer takes 3 hours each day for an uninterrupted 3 weeks then makes a Handle Animal skill check to train the wyvern for the general purpose combat riding (DC 67). Success means that if the trainer takes another 3 hours each day for another uninterrupted 3 weeks that the wyvern will gain the general purpose combat riding. Failure means that the trainer must start anew if a trained wyvern is still desired.
Let me explain that DC 67. According to the Player's Handbook, Dungeon Master's Guide, and Monster Manual, there's really no way to train a wyvern. The Dungeon Master's Guide would have a dude—a really persuasive dude—convince a wyvern to serve as his mount then, despite the absence of rules for doing so, have that same dude and wyvern train together… never explaining what that means.
So, if a trainer lacks the feat Dragon Trainer (Races of the Dragon 98-9)—which makes this DC 20—, a trainer must resort to the Epic Level Handbook's rules for Handle Animal (41), which set the DC for "[t]rain other creature" to 60 plus the creature's Hit Dice. And, while those rules are included in the System Resource Document, they weren't updated by the 3.5 revision.
(Further, the Epic Level Handbook sets the time required to train the other creature to 2 months, which a DM may fairly rule increases the time it takes to train a wyvern for the purpose combat riding to one year. That is, it normally takes 1 week to teach a creature a trick but 2 months an other creature… so if every week is instead 2 months, 6 weeks for the general purpose combat riding works out to a modified 12 months. In my experience, rarely does an adventurer enjoy this amount of time off.)
What training a wyvern yields
The Dungeon Master's Guide is unclear on why exactly a wyvern mount must be trained. The Rules of the Game Web Column "All about Mounts, Part Five" on Intelligent Mounts says
According to the Dungeon Master's Guide, a mount with an Intelligence score of 3 or higher acts like an NPC ally rather than a mount. Riding such an ally works much like riding an aggressive mount in battle (see Part Two)…
It's 5 or higher, but we'll let that slide… and, anyway, I tend to agree with Williams here: the cutoff shouldn't be Intelligence 5 or higher as per the DMG but 3 or higher, the cutoff between just instinct and what we consider intelligence. Anyway, so you know, "All about Mounts, Part 2" on "Aggressive Mounts in a Battle" says
A character could ride a mount that isn't cowardly in battle, but still isn't trained to carry a rider into combat.… If you find yourself in a battle while still astride your mount/ally, you and your improvised mount still act on your initiative number. You must attempt a Ride check to direct the mount's actions, which is noted in the section on unruly mounts. [That section says that, in short, the rider takes a move action that provokes attacks of opportunity to makes a Ride skill check (DC 20). Success means that the rider directs the mount.] If you fail, the mount might stick around to fight. If so, the mount moves where it will, but you still can't take any other action in the same round that you made the failed check (you spend your time just staying on).
On Intelligent Mounts continues, saying
…except that you don't have to make a Ride check to act while riding. (If your mount carries you willingly, the ride is smooth enough so your actions aren't restricted.) [This is, I think, supposed to be the Ride skill check fight with warhorse (DC 10) that, if failed, means that either rider or mount can attack but not both.] You also cannot make a Ride check to control the mount's actions, but you can make a Diplomacy (or possibly a Wild Empathy) check to get the mount to accept your direction. If you do, your mount acts just like a mount trained for combat riding.
Even if you cannot (or do not choose to) direct your mount's actions, you still can ride along, making your own initiative roll and possibly delaying until after your mount acts….
(Emphasis mine. Note that I reference the Rules of the Game articles here—despite controversy—because other information on mounts is largely unavailable in the dnd-3.5e corpus. While dnd-3e tended to treat even intelligent mounts like uncomfortably acquiescent slaves, dnd-3.5e's more nuanced view tends to treat intelligent mounts like the intelligent and independent creatures they are, and this series of articles in the only place I've found that details the mechanics of that relationship.)
So, in short, after spending 6 weeks to a year training the wyvern for combat riding, it seems as if the wyvern still keeps its limited wits about it and still tends to do what it wants unless its rider convinces it otherwise, making training an intelligent mount at all, ultimately, a pointless exercise.
The wyvern's loyalty
The wyvern is an NPC. A rider that wishes to earn its loyalty earns it the same way she would any other NPC's loyalty—a sheep costs 2 gp (PH 112)—, and rearing a baby wyvern from infancy will probably make it more loyal than normal. Exactly how the wyvern reacts to the rider's overtures is up to the DM.
The wyvern's feats
As an NPC, the DM designs the creature, picking its feats and skills and, if it possesses the nonelite or elite arrays, assigning its ability scores. If it gains class levels, the DM picks those, too. A creature that raises a wyvern typically has no say in the wyvern's choices.
1 The Intelligence 10 giant eagle (Monster Manual 93), Intelligence 5 griffon (139-40), Intelligence 6 howler (Int 6) (154-5), Intelligence 10 giant owl (205), and Intelligence 10 pegasus all require training before they'll bear a rider into combat. What happens when a rider tries to ride them into combat anyway is unclear, but I suspect they—through no fault of their own—behave like Aggressive Mounts in a Battle (see above).
You may also be interested in this question about quickly training creatures, this question about training nonanimals that possess Intelligence scores of 1 or 2, and this pathfinder question that discusses for dnd-3.5e the history of the difference between domesticated and nondomesticated animals.