I could travel with the first character to the FFXIV world of my second character?

I made a character on one world and then was told I can only have one character per world (is this a trial only thing?). After I made the character on the second world, I was in a new city, a new FFXIV Realm Reborn, and everything was different. Is there a way I could travel with the first character to the world of my second character? Is each world a separate "planet" with no travel in between?

Welcome to Eorzea! 

Free trial accounts indeed are restricted to only having one character per 'world' - this is also the same for full accounts that are on the 'Entry' level subscription rate (the cheapest subscription). To be able to have multiple characters on the one server you need to have at least a 'Standard' subscription which is more expensive (the 'legacy' subscription also allows multiple characters per world, but sadly you can't qualify for this as it's no longer offered).

Technically the term 'world' is a misnomer - it's a fancy name for a server. And yes, each server is separate. Although you cannot freely travel between servers, once you upgrade your account to a subscription, you will be able to make use of the Character Transfer Service which allows you to move a character from one server to another. However this requires a once only fee to be paid (separate to your subscription), but the trade off is the move is virtually instantaneous. 

As for your second character having started in a different city, this is because what city you start the game in is determined by what class you choose when you create your character (specifically, the choice of city is the location of that class's 'Guild'). Not long after starting you will be able to unlock the Armoury System which allows you to change your classes, so if you want to start in a specific city you will need to choose a class whose Guild is located there. The starting cities and which classes are associated with them are below:

Limsa Lominsa. Port city of pirates. Choosing Marauder or Arcanist will start you here.

Gridania. Forest city of nature. Choosing Conjurer, Archer or Lancer will start you off here.

Ul'dah. Desert city of merchants and finance. Choosing Thaumaturge, Gladiator or Pugilist will start you here. 

As well as being able to change class reasonably early in the game, as you progress through the story you will also open up travel between each city once you gain your airship pass so you'll eventually get to see all three cities without having to create a character in each.