which is best for fast switching do I need to use MOSFET driver IC such as ICL766 with IRFZ44n or is it better to use lower VGS MOSFET such as AO3400
You don't want fast switching to drive LED strips! If the FET switches in a couple tens of nanoseconds, you will send very high di/dt currents into your wires which will radiate electromagnetic interference and essentially act like a broadband radio jammer. Remember a signal with 10ns edges has bandwidth extending to hundreds of MHz...
On the contrary, you want slow edges, like 0.5µs - 1µs. This increases switching losses, but your PWM frequency is going to be low anyway, so switching losses, which are proportional to frequency, will be low too. Let's check the math:
Conduction losses = RdsON * I^2 * DutyCycle
For 2 amps and a duty cycle of 1 (100%) a FET with RdsON of 40 mOhms dissipates 0.16W -- you can use a FET with lower RdsON if you want.
Switching losses = V * I * SwitchingTime * Frequency
For 12V, 2A, 500ns, 10kHz losses will be 0.12W.
Note these losses are already a bit high for a SOT23 FET. I'd use a SO8 single or dual FET instead, they have better dissipation and they're pretty cheap.
Say you want to switch an AO3400 in T=500ns, total gate charge is Qg=7nC so gate drive current will be roughly Qg/T = 14mA. For this you don't need a specialized gate drive IC.
If you have 5V available, you can use a 74HCT logic gate as voltage translator between your 3V3 micro and a 5V "logic level" MOSFET. It's cheap and it'll work.
If you only have 3V3 available and no 5V, and you don't want to bother with a separate 5V supply, then it makes sense to use a FET that is compatible with 3V3 drive. Depending on the gate drive current required, and the output drive capability of your micro, you might want to add a logic gate as a buffer too. 74LVC for example.