This will be a long post, but my husband and I are experienced flyers and I promise it'll have loads of hopefully helpful info!
First, you have to factor in what airline you're flying on and what the route is. For example, I'm from the US and within the United States most "First Class" routes are simply in the front of the plane with wider seats that recline more, the food is better (and free) and you get extra perks like more bags checked free and often lounge access (which I typically already get since I'm BA Gold and United Gold). A handful of premium routes, such as NYC to LA will have some planes with the lie flat big seats that you usually see on international business class. This would also apply for the difference between BA economy and Club Europe on most routes as well.
My recommendation is to research premium economy, especially if you're travelling with children. Sometimes it's just a couple hundred more than economy and you get a wider seat, better food, a more private mini cabin, a good 6-8 inches more legroom. I recommend looking on SeatGuru.com to check the layout of the seats and see if you can request more favourable ones.
First and Business Class aren't the same, but again, it depends on the airline. For example, as others have mentioned Singapore and Emirates have a mindblowingly luxurious First product. For some other airlines, such as Qatar, their new Q suites business product is better than many international First cabins.
My husband and I will often use points from credit cards to upgrade. There are some excellent blogs dedicated to this for future reference, but if you haven't started collecting points and miles it might not work for this upcoming flight, plus not all economy fares (typically the cheapest) will be upgradeable to Premium Economy or Business.
My recommendation (especially if you're flying with BA) is to look into Premium Economy if you're flying on a route that offers it, especially if it's a newer plane. It might only cost a bit more for the upgrade. You will definitely get more comfort for a longer flight.
I'm happy to answer more questions if I'm able to.