There are several ways to start a kill streak but the easiest way for most people is to wait for an attack and then counter it (right trigger + X). If you want to provoke the guards into attack you then press the Taunt button (Y).
In order to carry out a successful streak, you can't be interrupted. Things that'll interrupt the streak:
1. Sand thrown at you (these can be dodged).
2. Countering an attack during a streak (this will start a new streak)
3. Dodging an attack prematurely.
4. Pausing too long.
5. Getting hit by any attacks (ranged or melee).
It seems like you don't know how to counter-grab and dodge. When a guard grabs you, you need to quickly break free of them. To do that, press right trigger and B. If no guards try to attack you, then continue the streak like you normally would. Getting grabbed then breaking free does not interrupt the streak.
However, if another guard does try to attack you after breaking free... You need to make sure your streak isn't interrupted by timing their attack and dodging it (right trigger + A). Once you've successfully dodged, just select another target with the left joystick and then press X (like you normally would). Your streak will continue.
If you find this too hard, then one way is to use the special execution with the sword/pistol or dagger/throwing daggers combinations.
Start a fight like you normally would, kill a guard with a counter or combo. Then while Ezio is finishing off that guard with a finishing move, select another guard with the left stick and HOLD down the X. Ezio will kill one guard after another in one move (for a total of two guards). During this time, no guards will try to attack you or grab you. As soon as you see him do this, let go of X, then select a new target and hold X down again. Rinse and repeat. It's very similar to executing a regular kill streak, except instead of pressing attack, you hold it down.
I probably confused you more than I helped you. So if none of this helped, then I suggest practicing in VR or just go around Rome and keep practicing. Good luck.