Instant glue works, but if you want to avoid it looking horrible you need to be really careful with how you apply it. The best option depends on the type of piece. For actual model plastic, it's worth using specific plastic cement (like Tamiya or similar), which slightly dissolves the surfaces and creates a chemical bond that's way stronger than cyanoacrylate. Cyanoacrylate is fine for small things or details, but it tends to leave visible marks and isn't ideal when you want the work to look clean on larger pieces.
What I'd do is get model plastic cement from a hobby shop or online, apply it with a toothpick or fine brush to control the amount, and then hold the pieces together with masking tape while it dries. Let it dry for the time the package says, don't try to rush it. If you use cyanoacrylate anyway, apply really tiny drops, wait a few seconds for it to sink in a bit, and then press. The difference is in not using too much glue, because that's what produces that ugly white mark you see.
One tip I'll give you: before gluing large pieces together, test the cement you choose on a couple of leftover plastic bits from the same kit to see how it looks when it's dry. That way you know what to expect and you won't ruin your model.