Some Raags move freely and some have a melodic motif which is established through the idea of Aaroh and Avroh (ascending and descending).

Let’s say I have 7 notes, and while ascending I skip the 3rd and while descending I skip the 5th.
I can play Do-Re-Mi, but I can’t continue from Mi (3rd) to Fa. So if i want to continue the scale i will play Do Re Fa Sol etc… or Do Re Mi Re Fa Sol etc…

This is a general approach off course and some Raags have exceptions and more complexed movements but this is the basic idea of Aaroh and Avroh.

the common movements one has to know and use frequently so the color or mood of the Raag can be noticed.

which are exercise used to assimilate the ascending-descending of the Raag as well as the common phrases.
I wrote those for students based on the exercises i got from my teachers. I usually memorized them and practiced for hours but never wrote them down.
So this is not an “official” exercise but something each musician can create for himself.

called “Bandish” “Cheez” or “Gat”.
As you already know (i hope) Indian Music is mostly improvised. But the composition is the base. It is what we play first after the Alap, and it is what we play when the Tabla player improvises. 
When we improvise the Tabla keeps a composition as well called “Teka” which helps us to understand where we are in the rhythmical cycle.

Improvising in a rhythmical cycle.
As opposed to western music  where the rhythmical concept is linear, in Indian Music it’s a cycle. 

It’s a bit hard to explain but imagine a clock that has 12 hours and when you pass the last hour you get to the first hour of the next cycle again. 
It’s the same idea as the Indian Rhythmical cycle called “Tala”. 

Improvising on a rhythmical cycle is rather difficult at the beginning so I created this system where you improvise each time on a very small portion of the cycle and gradually increase the space you have for improvisation.
You can follow the phrases I wrote for a start but I recommend to try and write your own improvisations based on mine. In that way, you will understand better the concept behind rhythmical improvisation.

A Tihai is a phrase that repeats 3 times and its last note falls on the first beat of the cycle (Sam) or the beginning of the composition (Mukra). 

In the video below, you can learn the idea of a Tihai, and I prepared a Tihai table as well where you can see examples of Tihais in different cycles. 
There are many attitudes for memorizing tihais and all of them are good. 
My approach is to always memorize the rhythmical pattern of a Tihai and then i can use it on any Raag I play.