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.
Above: Fire Breath in action
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.
Above: Dragonrend will be your favorite once you get it
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.