Any general tips, advice, or anything I can get to make my experience as grand as possible?
A gnome fighter, you say? I'd recommend one of two paths, if the halfling outrider, cavalier and whatever other mounty prestige classes aren't available from the complete books.
If you can play lawful good "Dudley Do-Right" do so. Take a level of cleric and worship a lawful/good or lawful good deity and go the path of the prestige paladin (XP penalty, sure, but you did pick a gnome fighter specializing in saddle-beasts, I think this might come with the territory). This will beef up your mount, martial ability, cleric spellcasting ability and not overwhelm you with a large array of quirky special abilities as compared to many other prestige classes. Basically they reworked the paladin into a class with requirements (also did this with bard and ranger) in the Unearthed Arcana rules. There are even paladin alignment variants in case being the "I'll save you Penelope!" type isn't exactly your style. Each still beefs up your mount and can scale with this prestige class option.
If you can play evil I highly recommend trying out the blackguard. They get an ability much like the paladin's mount, in fact they are much like the paladin at their core except for the fact that they're really wicked and require you to be evil. An advantage here is that they are core - straight from the Dungeon Master's Guide. A bit tougher to qualify for feat-wise, but their mount levels based on character level instead of class level. I once houseruled these as a neutral prestige class and it seemed to work well enough for our purposes, maybe your DM could do the same. For the blackguard, the trickiest part would be having five ranks of hide and peaceful contact with a summoned evil outsider.
Perhaps not the greatest or strongest way to manage your mount, but these options are some tried-and-true ways to have fun with it. Aside from class recommendation, I highly suggest the feat spirited charge. Even without a lance, that's a grand charge. Aside from that, see if your DM will allow the leadership feat so you can attract a cohort who can serve as a mount. I like that you want to keep your riding dog as long as possible (instead of treating this NPC companion as a means to an end like some, ugh), but unless your DM will lift the maximum hit dice cap for him he's not going to keep up for much longer if you don't bring in class abilities or something like to augment him.
That is a lot of questions, actually.
Fly
PFSRD on the matter:
You generally need only make a Fly check when you are attempting a complex maneuver. Without making a check, a flying creature can remain flying at the end of its turn so long as it moves a distance greater than half its speed.
The complex maneuver table there lists the actions possible (fly slowly, hover, turn sharply, ascend steeply). Less drastic maneuvers should be generally possible without a check. Please also note, that it gets a modifier to the Fly check from its maneuverability and size.
The Wingover feat allows it to turn 180 degrees for free once per turn, as opposed to expending movement and making a Fly check.
Attack
Talons are claws on the bird's feet. They are accessible when it is airborne; they are even accessible when it is standing on the ground (jumping a bit with a bit of help from wings).
It can attack 3 times if it takes a Full attack action; it is possible maintain an airborne position in that case with a successful Fly check to hover (since Fly checks do not consume an action). Please note that this method is not dependant on Wingover feat.
Without the Multiattack it does get a -5 modifier to all secondary attacks.
Which attack is primary or secondary depends on the attack type, and is indicated in the Universal Monster Rules for Natural Attacks. Bite, claw, gore, slam, sting and talon attacks are always primary. Other attacks can be primary if a creature doesn't have any of the aforementioned types. Primary attacks use the full Str modifier to damage (or 1.5x if the creature only has a single natural attack, or for special cases), secondary attacks get only 0.5x Str to damage, and are made at a -5 modifier (or -2 with the Multiattack feat).
For the Roc, all listed attacks are primary and get full Str to damage. The Multiattack feat has no effect since it only affects secondary attacks.
Tricks
Simple movement does not trigger a Fly check; a steep ascend and a sharp turn, on the other hand, does.
The attack of opportunity is triggered by many things, but in relation to movement it is triggered when someone leaves a threatened square. So if it initially was not in anyone's reach, an optimal movement to flanking position probably won't trigger any AoO's, since it will only enter a threatened space.
Maintaining a flanking position, given the opponent does not move, is a matter of a Fly check to hover (or standing on the ground), so no AoO's here.
Best Answer
Before anything else, remember that the animal companion is an NPC, and, while most GMs allow the player to build the animal companion, the GM needn't allow this, instead doing all the work himself and, if exceptionally Gygaxian, even keeping the animal companion's statistics secret from the player. The only thing the animal companion's master picks is the animal companion's bonus tricks (even ability score increases are picked by the animal companion!). Bear that in mind when considering what abilities the (not your) animal companion may have.
Anyway, this low-level advice assumes that the winged marauder alchemist starts at level 1 and that the player gets to build his character's animal companion.
Ride the giant vulture, command the dire bat
A player must consider two things when dealing with his character's flying animal companion:
What's the mount's carrying capacity? A creature with a load greater than light "counts [that load] as medium or heavy armor for the purpose of abilities or skills that are restricted by armor," and "[f]lying mounts can't fly in medium or heavy barding." The dire bat starts with Strength 9, its light load 30 lbs. The giant vulture starts with Strength 12, its light load 43 lbs. The typical female goblin weighs at least 27 lbs buck naked.1
Assuming the GM says that barding is armor (and unless he's an actual lawyer he probably will), the giant vulture may be able carry the skinny, naked female goblin and some of her gear and still fly.2 The dire bat probably can't.
How quickly can the creature get a +14 bonus to its Fly skill? Tactically, and for the real-life sanity of all involved, the creature will need to hover. A lot. That means the creature's skill ranks all go into Fly until that skill's at +14.
This is because it's unclear what happens when a winged creature tries to hover and fails the Fly skill check by 1–4. Getting that Fly to +14 quickly, then, means fewer arguments at the table.
Teach the animal companion fight and flee tricks; its feats will suck
Convince the DM that, before play begins, the character taught the animal companion all the tricks he could. The animal companion will die anyway, so it won't matter beyond, like, fight 4, but that's when the character'll need all the tricks he can muster from his companion. Pushing the animal is hard at low levels. That means, one hopes, 7 tricks.
Take the attack trick twice, defend, down, flee, heel, and stay. When it dies, pick attack as the bonus trick. If the character has 6 weeks, the special purpose combat training (or, when its carrying capacity allows, a war saddle (5,000 gp; 20 lbs.)) and the attack trick again as the bonus trick will meet most needs.
The animal companion's feats will be horrible to start, limited as they are to small list of feats until the character can allocate an ability score increase to grant it Intelligence 3. The feat Skill Focus (Fly) is a serious option, as is the feat Weapon Finesse. The feat Flyby Attack isn't; it's not on the list.
Take feats to make bombs better not the animal companion
Better bombs mean more dead enemies faster and a safer animal companion. Beef up bombs with feats to save the animal companion.
That said, if riding the giant vulture the character should have the feat Mounted Combat. It's a gateway feat, gives a low-level alchemist something to do with his immediate action, and may save the animal companion's life.3 After that, Mounted Archery is a possibility, but the goblin alchemist is making ranged touch attacks with his bombs so if it's not needed, don't take it. I like the idea of the feat Indomitable Mount, but I've not used it.
The alchemist is probably commanding the dire bat to engage in combat or employing it for utility, so the feats Evolved Companion (maybe picking bleed, claws, or reach) and Totem Beast are acceptable, but I prefer the feat Spirit's Gift, which although requiring some research is extremely versatile.
As the alchemist doesn't technically cast spells, the character's forbidden from taking the best teamwork feat, Improved Spell Sharing.4 Other teamwork feats can be taken, but most are a waste.5
Don't plan on the same companion seeing later levels
The alchemist will probably replace an Intelligence 3 animal companion whenever the creature gains a new feat so he can change the creature's entire feat load so the creature can do better, cooler things.
Eventually, bonus tricks will and a high Handle Animal skill bonus for pushing the animal companion will be sufficient to make the Intelligence 3 animal companion a valuable addition to the party. At low levels, though, a flying animal companion that's intended as a mount is interesting but difficult and fragile.
1 Fat goblins don't ride flying mounts. Well, they do, but the mounts can't fly when they do. So they're less mounts that can fly and more creatures that could fly if this fat goblin weren't on my back. Goblins probably think this is hilarious.
2 Even a Small peasant's outfit weighs 0.5 lbs.
3 In the long run, the character won't care. In the short run, when he's 1,000 ft. up and the flying companion he's riding dies, he suddenly will. I think many goblins who ride flying creatures' last words are, "I knew I should've taken Mounted Combat."
4 Not the similarly named Improved Share Spells. Sigh. Pathfinder.
5 Okay, Shake It Off has saved my inquisitor's life a few times. Also, a mount with the horsemaster's saddle (12,000 gp; 5 lbs.) or, better yet, the sky marines elite saddle (14,000 gp; 5 lbs.) shares its rider's teamwork feats.