Neighborhoods are used to increase housing in the mid-to-late game, based on the appeal of the tile they are. You can build multiple Neighborhoods in a city. That means any housing problem gets solved with them (except if you have a cramped city). It's quite a no-brainer once you see them in action (and understand what might boost or reduce appeal to maximise its effect).
Lastly, about trade routes. Internal trade routes are very powerful, and a key element to playing wide. Send your trade routes from your newly-founded cities to your core cities (e.g. the ones with many districts). Every district in the destination gives either +1 food or +1 production to the other city. 3-4 routes should jump-start your new city and make it productive faster. Also, the policy cards that gives gold for every trade route and some wonders such as Great Zimbabwe also work, boosting your gold even further.