According to this question, Dragons are vulnerable to different types of damage:
However, the answer only advocates seeing whether they have fire or ice breath to know. Is there a way I can tell before I am fully engaged with them (e.g. by appearance), and are there any types other than the fire or ice breathing kind that I can detect?
According to the wiki yes, there is a way to tell by their names:
Red dragons: These dragons have appeared in Skyrim after the emergence of Alduin. Their bodies have a greyish tone with a light grey underbelly. Yol (fire) is their breath attack.
Green dragons: These Dragons have appeared in Skyrim after the emergence of Alduin. They have black skin with a distinct green glow. Commonly found in forested areas, they can use the Raan Shout (animal alligiance) to call nearby animals to their aid.
Frost dragons: Frost dragons appear in Skyrim after the emergence of Alduin. Frost dragons are white with massive protruding black spikes coming out of their back. They predominatly use Fo (frost) breath that deals damage, slows the player down, and reduces stamina.
Blood dragons: Blood dragons appear in Skyrim after the emergence of Alduin. They are recognized by their green-grey, crowned head, and large fan formation on the end of their tail. They become common at level 15-16, and use Yol (fire) and Fo (frost) breath attacks.
Elder dragons: Elder dragons appear in Skyrim after the emergence of Alduin. Orangeish in color can use either Yol (fire) or Fo (frost). These attacks are generally strong and debilitating. Their tails can rend a forcefull hit and have the spade shape at the tip.
Ancient dragons: Ancient dragons appear in Skyrim after the emergence of Alduin. These dragons have a tremendous amount of health and can use either Yol (fire) or Fo (frost) as the Elder Dragons do. Just as Elder Dragons they have a spade shaped tail tip.
Dragons that use fire based attacks are 50% more resistant to fire based attacks although 25% weaker when faced with frost based attacks. Vice-versa for frost breathing dragons
Using those clues, you can try to discern what type they are before you are fully engaged but most dragons seem to be of either type, so there's no easy way to tell unless you spot them fighting something else.
There are several ways to figure out what type of dragon you are looking at(fire breathing or frost breathing). The easiest way is on the PC. Use the tilde key to bring up the developer's console, click on the dragon and type getav fireresist followed by the enter key. If it's a fire breather you'll see a value of 50.00 and if it is a frost breather a value of -25.00 will be displayed. Hit the tilde key again to close the developer's console and resume play.
If you're on a console(or on PC and don't want to use the developer's console) there are still a couple ways to make this determination. The first is to alternately cast fire and frost spells of the same level at the dragon and see which does more damage to its health bar(difficult if it's flying around). The disparity in damage should be pretty obvious(resistance to your element means it takes half damage, weakness means it takes 25% more damage than normal or 2.5 times the damage compared to damage from the element of which it's resistant).
The final way is to wait for it to shout its element at you. This can be a bit of a waiting game at very high player level since most dragons you face will be the legendary type. They tend to prefer their Drain Vitality shout over their elemental shout and will probably use Drain Vitality several times before they use their other shout. Once their element comes out of their mouth, you'll know.
