I was just having a discussion with a brother who said that a spouse cheating on their spouse isn't haram? And that lying to your wife isn't haram too?

He also said that a husband would not need to take the permission of his first wife if he's thinking about marrying another woman as his second.

What are the rulings with these things?

Cheating and lying are sins, so they are both definitely haraam.

Whilst husband doesn’t not need permission of his first wife from a shari point of view, but if it is illegal to marry a second in any country then it is not allowed, and if in the law of the country one needs permission of the first wife, then it would be necessary.