First off, learn the fast q combo first. It is the most essential riven combo you need to learn, and will increase your damage by a lot.
Here's a video explaining how to do it. Play Video Basic Combos
Probably the most simple animation cancel is E-R. For this, press E and then use R right after. This lets you cancel the animation of R, which self-stuns you for about a second, and merges it with your E animation. Basically, this lets you be able to use R when chasing or mid fight without having to be self-stunned or stop moving.
Another version of this is R-W, which lets you use both skills and only be self-stunned for the duration of W. Since they both stop you from moving, this only causes you to stop in place for 1 second instead of 2 if you used them seperately.
R2 - Q lets you cancel the animation of R2 while using q. If used correctly, Q will hit BEFORE R2, which causes R2 to do more damage as it is an execute. Useful for burst at the end or middle of a combo.

Lane Combos
Q - W - AA and use E and Q to get out. This pokes them without much risk of you taking damage. Does not work against certain matchups like renekton, who can run you down with e and trade back.
All in combo : E - R - AA - Q - AA - Q - W - AA - R2 - Q
There are many variations of all in, just make sure that you auto between Q's.

Doublecasting
Next up is the one I personally like the best, the doublecast. Doublecasting is a mechanic where after you e, there is a small time frame where you can cancel any skill animation by using the ability and then using q right after. For example, E - W - Q would cause w and q to be used at the exact same time. Here is an example from Adrian Riven, a high elo NA riven main. Play Video By learning this, you can do a lot of burst damage in one frame. (unknown embed type)In this clip, the riven in the video does Flash - E - Autoattack - W - Q, which procs thunderlords and surprises the jax with the damage, leaving almost no reaction possible.
Oneshot Combos