What effect will it have if I put a bigger sprocket on my bicycle?

First you need to be clear about what you're talking about.

In normal bike-speak the sprocket(s) go at the rear wheel.

Either only one, or a stack of them in the shape of a freewheel or a cassette.

Up by the pedals you have the chainwheel(s) / chainrings(s).

Now, for some strange reason, often in BMX-speak, the sprocket goes at the front, and what you have at the back is called a driver.

But let's stick to regular bike-speak, sprocket = rear, chainring = front.

In this case, going to a bigger sprocket will make the bike easier to pedal.

You get more power, but less speed.

The rear wheel will turn less for each turn of the pedals. Good for getting going, bad for going fast.

Going to a bigger chainring will have the opposite result.

More speed, less power.

You'll need to push hard to get the bike started, but your top speed can be higher. The rear wheel will turn more for each turn of the pedals.

If you get both a bigger chainring and a bigger sprocket, the changes will kinda-sorta cancel each other out. Not exactly, unless you get them both proportionally equally bigger. But pretty much.

Let's the tooth count is 12 rear and 36 front, and you change to 14 rear 42 front.

This time nothing really changes as 36/12=3 and 42/14=3. The same number of turns at the wheel for each turn of the pedals.