If I trade a player with a multi-year contract to another team, is that team now responsible for the bonus money payout or am I still responsible for the whole bonus amount??

You are. All unearned bonus money becomes a dead cap hit.

Honestly, there should be a pop up box showing the cap implications similar to what shows when you are about to cut a player.

Correct...the team a player is traded to is only responsible for his yearly base salary and none of the bonus.

The bonus creates a dead cap hit to the team that traded him. With big enough contracts (6 year/80 million) that cap hit can be huge which can ruin teams

To expand on this answer:
The bonus for the current year counts as dead cap this year.
The bonus for ALL remaining years are totalled and count as dead cap ALL in the next year.

Ex:
A player has a 6 year contract with $5 million bonus each year ($30 million total bonus).
If you trade this player you take a $5 million dead cap hit this year, and a $25 million dead cap hit next year.

Just look at it the legal way: The player has signed a contract including a signing bonus. The bonus is paid out immediately. For financial balances (salary cap) the bonus will be split over the duration of the contract. If you trade the player, you cannot split the bonus any longer and the rest will be summed up as dead cap the following year.

Yikes!! Good thing I didn't trade the player. Good info to know next time I decide to trade.