The basic difficulty is that building a new city to collect a new resource doesn't make your people happy. The extra city eats up all the extra happiness and just gives you an extra city that you need to work, defend, and grow.
But if you have that luxury resource in the radius of your new city (or you have enough gold to buy a few tiles), you should only be what, like 4-7 turns away from +5 Happiness from that luxury? I know you're literally correct that more cities and more population = more unhappiness, but assuming you place your cities near at least one luxury (in the very early game), that shouldn't be a problem for long at all.
I also manage my overall expansion more carefully in Civ 5... I don't build Settler after Settler in the early game in a land-grab like I used to. I usually get 3 or maybe 4 cities up and running and then chill for a while, improving tiles, adding buildings, connecting trade routes, accumulating (and spending) culture, gaining city-state influence for more luxuries, trading with the AI for cash or more luxuries, etc. After those initial 3-4 cities are doing well, I add a couple more and repeat the process.
I wouldn't say the game punishes all expansion... but it does punish rapid, indiscriminate expansion.
Yeah,i tried both settling near new luxury resources and trading with other civs. My problem is,i can get a boost from 2-3 luxury resources,but after that,i cant find any new resource to boost my happiness.
And i dont know if it just me,but trading with other civs is not working,i ask for ONE resource and they want THREE + money O_O As for city states,to get the resource you need to be friends with them,and i have yet to figure out an efficent way to keep them friend to me.
The only way i found to gain some influence is to kill barbs,i am failing at every other way(they want GP,they want a luxury resource,they want me to conquer another city state.....too much for my poor growing empire),and bribing is not an option,costs too much. :-(
1) Are you making sure you improve the tiles with the luxury resources near your cities? If they're not in your immediate city radius, are you buying tiles to bring the luxury within your borders, and then improving that luxury tile? Are you directing your early-game research toward techs that enable you to build those improvements, like Calendar to build Plantations?
2) If befriending city-states and trading with other great powers is problematic, try and look closer to home to resolve your happiness problems. Make sure you don't neglect social policies - the Liberty tree can help, and one of the Honor tree abilities can reduce unhappiness as well.
3) If you're having money problems early on - are you being careful about adding new buildings and creating new units? Maintenance costs can be a huge dead weight if you build too much too fast. If you're selective about your buildings and only create military when you need it, cash shouldn't be so hard. Also, don't necessarily garrison every single city all the time - since cities can protect themselves, the old days of 'ungarrisoned city = barbarian city' are gone.
4) If you can't trade with the AI for their luxuries, sell them your extras. You should be able to get a few hundred gold for your second source of gems, dyes, furs, etc. OH! Be sure you're not trading away your ONLY source of a given luxury too - that took me by surprise in my first game, that I my only source of Pearls was available for trade. I thought only extras showed up in the available trade window.
Hope this helps, good luck!