I'm looking at tickets right now, $735 for economy, $3500 - $6000+ for business, $15,000+ for first class. Fortunately I have a lot of miles, so a 160K mile business class awards ticket is a possibility.
That's still out of reach of many people.
Put it off until you can pay for a premium experience which you'll enjoy more
I've vacationed in economy, business class, and first class, and the flight is the least memorable part of the trip -- I'd rather spend more money at the destination. For sure, a lay flat seat on a long-haul flight is fantastic compared to being strapped in a coach seat for 8+ hours, but unless I can pay for it with miles, I'd rather spend the money on a nicer hotel or longer stay (or just keep that money in my pocket)
And anyway even with the best possible credit card you have to be pretty rich to spend enough to earn enough points for a transatlantic business class flight for your family.