So I would like to ask since I just started Udyr, what is the first ability that I should max? I've seen many builds of pro players that max R first but I would like to know the reason. Why R and not Q since Q does more damage in battles? Just so you can have the faster clear time on camps?
It mainly depends on where you play him. In lane, taking Phoenix would be counterproductive as Tiger gives you better last-hit power and duel power, while Phoenix would just push your wave all the time, but I'll speak on behalf of Jungle Udyr:
The reason Phoenix is better in the jungle is - contrary to the popular belief that tiger stance gives better clear speed with it's synergy with the machete - it scales better with attack speed, and attack speed is what junglers want.
Phoenix gives you AOE damage on top of your AA damage, with the addition that every 3rd AA as well as the first one on activation will trigger a small AOE nuke. This gives Phoenix udyr higher item-less DPS, and encourages you to build attack speed to synergize with Feral Flare, instead of AD to make tiger stance's dot remotely useful. Besides that, you will deal more damage with Phoenix waves and AA procs than Tiger's DoT and AAs thanks to monkey's agility and Udyr's reactivation cooldown. This benefit also works for ganks assuming the lane has CC (in addition to red buff and monkey's agility MS bonus) as you will land a lot more attacks, while not even needing to attack for the phoenix activation flames to hit for some damage.
The difference in jungling speed between phoenix and tiger (up to the small wraith camp) is around 10 seconds, as the lack of AOE for clearing camps actually makes you lose a lot of time.
Both skill paths are possible. You can even Put early points into R and then switch to Q.
As you've already mentioned the Q is better for single target damage while R is alot faster when clearing camps. Most people max R nowadays because Q got nerfed mid season 3 and made the clearing even slower. If you're unexperienced with udyr I'd recommend maxing R over Q because it makes jungling a lot easier and it also makes you one of the fastest if not THE fastest jungler in the game. I would only max Q if you are planning to snowball extremely hard and if you can afford it. The damage of R is also significant during ganks and while it might not be as good as Q the cleartime boost really is worth maxing R.
Udyr on lane: I recommend to max Q, as, you mentionned it, deals more single target damage. Just farm in bear stance and when your opponent tries to go near you stun + q proc. R isn't really handy on lane because it will push hard, thus making you vulnerable to ganks and tower hits. You're nearly forced to max tiger in lane or you'll have a too hard time.
Udyr jungle: Max R, or at least lvl 3 (when udyr is lvl 5), the clearing times are indeed awesome, and the spell damage is great even without stuff. The q does incredible damage, yes, but only if you took some ad (triforce / brk / zephyr) and makes you an 1v1 beast. Problem: that's not your job. Your job as Udyr in teamfights is to run on the enemies and stun everybody, and then, deal damage. Another advantage about maxing R is when ganking: the phoenix damage is instant while q is a DoT, making it hard for your laner to take the kill (q + red + minions...). But as I said, it is also good to get R lvl 3 and then go on tiger, it all depends on the context: your team, the opponents', your playstyle, their playstyle etc... Keep in mind: q = duel + splitpush, r = ganks and teamfights.
The reason you don't see jungle Udyr being played much is because there are trade offs depending on what route you go - assuming you're facing decent opponents.
The ultimate problem is that Udyr can get kited. He needs to physically run towards the enemy. Other popular junglers like Nocturne, Kha, Vi, Elise, and Lee all have quick gap closers. If I build Wriggles/Feral Flare then by the time I reach the enemy to stun them, I'd have lost 1/2 of my HP. I might get CC'd by the enemy support and even get lose 2/3 of my HP. That's not good.
If I build Ancient Golem then I might only lose 1/3 of my HP and I'll have CC reduction from the tenacity. Then I'll be able to stun the enemy and tank for my team. That is a safer role which does not rely on the mistakes of the enemy team to get me the win.
If you end up building Wriggles you might as well max Q because you're going man-mode!! Hopefully you get early kills and snowball to the point where you can 1v1 anyone.
If you end up building Ancient Golem you might as well max R because you won't be blowing up the enemy carries anyway. The faster clear speed and lane pushing power is useful. You might consider building a Wits End in the mid game which gives you a mix of offensive and defensive stats.
Depends your playstyle.
I like Q because it still has a good clear speed but it is great for dueling and counter jungling.
R is more for faster camp clears and being in the jungle longer.
My play style isn't good with R at all so I max Q but my friend goes R because he likes farming more than dueling and ganking early.