How many types of inertia are there? I was looking for types of inertia, but I am confused. My book says there are three types of inertia, namely inertia of rest, inertia of motion, and inertia of direction. But when I searched for these types in good books like Halliday, Randall, Hewitt, Zemansky, no author has talked of types of inertia.
Now I feel as there are no types of mass, so there need not be types of inertia, as inertia is measure of mass.
Also, my book says that the earth going around the sun continuously is an example of inertia.
 A: Most generally inertia is resistance to change. This can apply to your daily life too. For example sleep inertia is when you wake up groggy and a bit slow and you'd rather go back to sleep. Your body resists changing from being asleep to not being asleep.
The three types of inertia mentioned by your book are vague statements about how objects resist changing being at rest, changing their motion and changing their direction of motion$^\dagger$. Here I don't mean vague in a negative way, it's just a more conceptual, less precise way of stating things. These three types of inertia are things you have probably observed in your daily life and it is a way of categorising those observations.
It turns out that all those three types are actually the same thing. There is only one type of inertia: inertia of motion. And it is caused by mass. Being at rest is the same as moving with zero velocity. Changing your motion is the same as changing your velocity. Changing direction of motion is likewise the same as changing your velocity.
So the fact that these types of inertia are the same can be seen as a 'result' from physics. You're probably so used to this concept that you can't tell the difference. Personally I wouldn't bother splitting inertia up into three parts and naming them. I have also never heard of this before so it might just be something the author came up with.
$\dagger$ I'm not sure if inertia of direction refers to bodies not wanting to change direction or if it refers to objects spinning really fast not wanting to change orientation. Either way it shouldn't influence my story.
A: It sounds like your book was written by Aristotle. According to Einstein, physics works the same in all inertial reference frames. So the amount of force needed to accelerate a moving object is the same as that for one "at rest", and accelerating an object tangentially to its direction of motion takes the same amount of force as an acceleration normal to the direction of motion; given any acceleration, for each of these categories, there is some frame reference such that the acceleration is in that category. For instance, if you're looking at the force needed to accelerate a moving car, there is some frame of reference where the car is at rest, and so what is "inertia of motion" in one reference frame would be "inertia of rest" in another.
The author of this book may be confusing friction and inertia. The force needed to get an object at rest moving is more than that needed to keep it moving, but that is friction, not inertia. I'm not aware of any third coefficient of friction for acceleration normal to the direction of motion.
A: My suggestion is you throw this book away, and use only "good books".
Whoever wrote this book has a very confused mind.
I can vaguely guess what the author might have had in mind when he mentioned these "three types", but trying to make sense of these very confused notions would only add to your confusion, not reduce it.
Also, while the revolution of the Earth around the Sun does involve inertia, the overall motion is certainly not exclusively due to inertia.
Just as a matter by curiosity (certainly not to buy it) what are the title and author of this book ?
EDIT
Maybe I should have given you more details earlier.
The inertia that, when no force is applied, keeps at rest a body at rest is the same that guarantees a body in motion "perseveres" in this motion at the same speed in the same direction. So what your book call "inertia of rest", "inertia of motion" (constant value of the speed in meters per second?) "inertia of direction" (which, combined with the previous one, is the only meaningful thing for a physicist, conservation of the "physical speed"  which means both direction and value in meters per second). Distinguishing them as three "types of inertia" makes absolutely no sense.
Now about the Earth and the Sun.
Before Copernicus (thank you, user4574), people believed the Sun went around the Earth. Now we know that it is the contrary, the Earth goes around the Sun. But this motion, called "revolution" takes a whole year and in combination with the tilt of the Earth's axis is responsible for the seasons. This motion does involve inertia, but also a force. It is the interaction of this force, the gravitation of the Sun, and the inertia of Earth that is responsible for it, not just some "rotational inertia" that by itself would keep the Earth to go around the Sun because it is now rotating around it.
There is really, however, such a thing as "rotational inertia". But not to go round another object: to keep an object rotating around itself (technically, around it own axis), like a rotating top, to keep rotating indefinitely if no (momentum of) force is applied on it. I am not going to elaborate on the notion of "momentum of force".
And indeed the Earth does rotate around its own axis. Not once a year but once a day. This is why we see the Sun rise and set every day. This daily motion does not mean that the Earth rotates around the Sun but rather rotates around its own axis, once a day.
And this daily motion, yes, is purely due to inertia, a different sort of inertia than the first one, technically called "conservation of angular momentum", but you can think about it, if you want, as a "rotational inertia" though to my knowledge nobody uses this term.
So yes, sunrise and sunset are phenomenons due to inertia, to the fact that the Earth keeps turning around its own axis, like a rotating top.
But it was not at all clear that you were referring to this daily motion in your question, since you were mentioning "going round the Sun", which I understood as the year-long revolution, which is not purely due to inertia.
A: About inertia:
Compare the following two cases:
-you start running from standstill
-from running you slow yourself down to standstill
In both cases a force is required to cause change of velocity. It takes effort to get up to a speed; it takes effort to bring speed back down.
As we know, we call this common factor 'inertia';  a force is required to cause change of velocity
One is easily tempted to suppose that there is also a difference between the two cases. It is tempting to suppose that acceleration and deceleration are in some real sense different.
But then:
Here is something that one can actually try:
Let's say you are in a train carriage, a train carriage with room enough to run along the length. (Yeah, an actual train is too cramped for that, but work with me here.)
Let the train ride be so smooth that you cannot guess the speed of the train relative to the ground. Let the windows be blinded, so you cannot look outside. Let that train be moving at a speed that corresponds to normal running speed.
So:
If you get up to running speed in that train carriage: it might be the case that the act of bringing yourself up to running speed has resulted in making yourself stationary with respect to the ground. There is no way of telling the difference.
Observations like that lead to the following generalization:
Inertia only tells you how much change of velocity there is, inertia never tells you how much velocity you have.
This is a rule to which no exceptions are known, for centuries now the physics community has been confident to think of it as a Law of physics:
Inertia only tells you how much change of velocity there is.
It doesn't get more fundamental than that.

According to you your book asserts three types of inertia. That assertion is in contradiction with one of the fundamental laws of motion.
The conclusion is inevitable: the author of that book must be very confused indeed.
A: Newton’s first law (Wikipedia)

Law 1. A body continues in its state of rest, or in uniform motion in a straight line, unless acted upon by a force.

Inertia of rest ist the same as inertia of motion when you consider the value 0 for speed like any other value of speed. Inertia of motion is the same as inertia of direction if you consider that motion always has a direction.
Or in other words: velocity is a vector; it has direction and length. The vector does not change “unless acted upon by force”
Thus, your text book created some unnecessary confusion by stating that there are three kinds of inertia while there actually is only one kind.
Even more confusing (and wrong, as @Alfred already said) is the example of earth’s motion around sun.
A: There aren’t three “types” of inertia. Inertia is a property of mass in which remains at rest or continues to move at constant speed in a straight line unless acted upon by a net force, per Newton’s first law.
It appears your book is simply describing these three aspects of the first law  ( rest, motion, direction). In my opinion the book shouldn’t refer to these as “types” of inertia.
Hope this helps
A: Use Newton's laws here
Inertia of rest: "An object at rest stays at rest, unless a net force acts on it"
Inertia of motion: "An object in motion stays in motion, unless a net force acts on it"
Inertia of direction: An object does not change direction on it's own without a force. Centripetal acceleration can change an object's direction. Take that you are in a car. You curve and feel that you are being pushed outwards. That's "centrifugal force", a nonexistent force caused by your inertia of direction, as you are not curving with the car, but moving tangent to the curve.
