There are four ways to approach dragon hunting: three based around each of the basic classes and one that can be considered the most ideal way of doing so. Every combat strategy shares a few major details, notably that any equipment or potions that let you resist physical, ice or fire damage are ideal things for you to bring along. Other than that your playing style will dictate how difficult these fights are and how you have to change up your tactics.
Possibly the most important thing about these fights is what sort of Shout you are going to use. The most effective Shouts in the game are:
Fire Breath: Useful against ice breathing dragons, notably the Frost Dragons.
Frost Breath: Useful against fire breathing dragons.
Marked for Death: This is immensely powerful against dragons when they land.
Dragonrend: You get this near the end of the storyline just as you’re battling dragons constantly. But since it forces dragons to land when they’re hit by it you’re going to be using this non-stop once it’s dragon-murder time.
There are other Shouts that can be useful depending on your playing style or the immediate situation but these are the most widely useful ones. For example, Become Ethereal or Slow Time can help if your hit points are getting low. If the dragon is picking a fight near a mammoth or sabre cat you can use Animal Allegiance to gain some assistance. Late in the game you gain the options to use Call Dragon or Call of Valor - these can be incredibly helpful but the recharge times make them prohibitive even in these situations.
Until you actually get Dragonrend you’ll need to draw the dragons into landing manually. Try to stay in wide open areas at all times. If you’re on a mountain path either run back to the ground or run up to a plateau to coerce the dragon into landing for melee combat. It won’t necessarily choose to land just because you’re out in the open - it may simply choose to keep strafing you with its breath, but this is the only way to get them to land of their own accord.
For the ideal setup you will want to have a strong bow with a decent enchantment, a strong melee weapon and either a shield or second weapon. Armor isn’t the most important so either light or heavy will be fine as long as you can move quickly in it. If you can carry a ring or amulet to help mitigate the damage from the breath weapon then you will be in very good shape to handle the hardest part of this encounter, since the parts when they’re flying around breathing on you can really drag down your health.
The weapon and shield can be replaced with dual wielding magic if you’re playing a magic user - just be sure that you aren’t using the same magic as the dragon's breath attack since it takes reduced damage from that source. Should you have invested in that magic type heavily, then it’s possible to use it against them, but don’t expect it to work the miracles it does on regular enemies. However, a magic user will still need to carry a bow (or use the Bound Bow spell) since the dragon moves around so much and will tend to be outside of spellcasting range.
Good weapon enchantments to have on your bow are elemental damage (although this forces you to use two bows for different dragon types), or Silent Moon Enchant. While the Silent Moon Enchant is only useful at night, the damage bonus (not fire, even though it’s described as “burning” the target) is very effective all around. The weapon enchantment should be Absorb Health since you’re going to lose it rapidly in melee combat with the dragon.
With this setup you can simply fire your bow at the dragon until it comes close enough for spells, shouts or melee attacks. Honestly, this build is simple to play and can handle just about any situation whether it’s a dragon or not. This is the build used in most battles in this guide.
In something of an inverse of what you might think, warriors have a hell of a time fighting dragons. If you’re playing a pure melee character without the benefit of spells or bows then you’ll have only your Shout attacks to battle the behemoths. As you might guess it’s not exactly peaches and rainbows when that dragon really wants to roast you alive from the sky without actually ever landing to battle you. That leaves you with one option, wait for it to land and…
A melee character is best off avoiding dealing with the dragons as much as he can until acquiring the Dragonrend shout. When used this reduces a dragon's defenses while also forcing it to land. This Shout is obtained by following the main storyline until completion of the quest Alduin’s Bane. Upon acquiring this Thu’um you can force the dragon to never even take off, which allows you to just run up to it with your weapon of choice to beat its face in.
The above may sound simple but until you get that ability every fight with a dragon is going to be a trial of patience if you don’t want to use a bow or magic. Your best bet is to have a follower who does use magic or a bow and then put yourself under cover. Step inside of a tent, under an awning or just out of view with your shield raised (if you have the perk Elemental Protection). Once the dragon lands, charge in and beat on it, preferably before it hurts you too much. Rinse and repeat to end the fight.
You may find yourself using your Shouts a lot to get ranged damage in. If this is the case try to get an Amulet of Talos as it reduces the cooldown on your Shouts by 20% which is a huge boon.
A pure melee thief setup isn’t going to have a whole lot of success dealing with dragons. The problem is that you can’t use sneak attacks on them and you’re probably using daggers, which don’t deal damage fast enough to kill dragons with any expedience. For these situations you’re going to have to use a powerful sword to make up the difference. The best thing to do is use a bow for dragon, mammoth and giant battles to gain experience enough with it to do serious damage to dragons with your arrows. Equip the strongest bow you can get, give it a good enchantment and then buy the best arrows available to deal maximum damage.
Even if you are sticking to using your bow for all encounters you might want to bring a shield to any dragon battles. Stick a Fortify Health enchantment onto the shield to enable yourself to stand toe to toe with the beast if it gets right in your face. However you’ll want to pull back constantly to try to get some space, then use the quick change menu to swap back to the bow.
Honestly a thief build should have some of the easiest dragon fights out there. The bow is an incredibly powerful weapon when trained up a bit and it can hit the dragon no matter where it hangs out. You might think a warrior wading into melee would have the easiest time but until you get Dragonrend that dragon may simply choose to never land. If that happens the archer can still yank down their health rapidly while a melee build stands there looking silly.
There are two distinct types of mages: the ones who run around in robes and use pure magic and the ones who use armor and weapons to complement their magical abilities. Both of them have a pretty easy time dealing with dragons so long as they’re given the right abilities to handle each dragon that comes after them.
The pure mage has a tough time since for the most part only destruction magic is helpful against dragons although summoning up an Atronach can help while the enemy is at a distance. Illusion just flat out doesn’t work without high level perks and Alteration is mostly useless. Having a decently leveled Destruction skill with the right perks will allow you to hammer on the dragons when they’re both at a distance and close by.
A good strategy to deal with dragons is as follows, using an ice dragon as the example: Summon up a Flame Atronach and start pummeling the dragon with fireballs (one-handed to preserve magicka) until it moves closer. Once it does so dual-wield the Flames spell to burn the living crap out of the beast, guzzling down potions to keep your magicka up until the beast either dies or flees back up into the air. Note that since lightning spells fire instantly, they're great for hitting flying dragons since you don't have to lead your shots, and they work for getting a dragon to land, too.
One of the safest ways to kill dragons with a mage character is if you’re using a battle mage type of character. Slap on a strong suit of armor that you’ve enchanted to enhance your survivability (resistances or health increase) then get a weapon that will drain HP from your enemies - a sword works best as it strikes fastest. Circle strafe around the dragon using your spell of choice while hitting the dragon with your sword. The dragon will often get confused and keep turning to keep up with you while you’re pulling his health down and keeping your own up.
Frankly a mage has an easy enough time so long as you can get a bunch of equipment on to bolster both your base magicka levels and regeneration rate.