Which capacitors to use with voltage regulator IC such as 7805?
Many of the tutorials suggest to use a large smoothing capacitor (tens and even hundreds of uF) with a full wave rectifier

An interesting property of AC voltage is that it goes down to zero twice per cycle, which means if you use a transformer and a rectifier, you need a capacitor large enough after the rectifier to act as temporary energy storage during the parts of the period where the AC voltage is too low.
Basically, \$ dv/dt = i/C \$ so you take the max post-rectifier voltage and the min allowed voltage at the input of the regulator, which gives you max allowed voltage drop, divide by one half period, you get your max dv/dt, you know the max i, you get the minimum C value to use. Add a wide safety factor.

but the datasheets of the voltage regulators and some tutorials suggest only a 0.33uF capacitor to be used for smoothing the input voltage.

That's to keep the regulator stable if you feed it from a DC power supply.
Regulators can react badly to too much inductance in the supply (ie, long wires) so a local small value cap keeps this power supply impedance down. Use whatever ceramic cap of suitable voltage like 100nF or 1µF, value doesn't matter as long as it's enough, what matters is that it is placed close to the regulator.

Moreover, what if we use a large capacitor such as 10uF instead of 0.1uF after the voltage regulator?

Some regulators are conditionally stable depending on output cap value and ESR, so reading the datasheet is necessary. 7805 isn't picky though. A larger output cap will probably just give better transient response.