Upgrading from 2007 or 2010 to 2013
There is no direct upgrade path. It’s a case of attaching the content databases to the new installation. However there are some tools such as Lightning Tools and ShareGate which will make the process easier. When you attach the new content databases they are displayed in 2010 format but can temporarily upgrade each site to 2013 format for 30days to allow testing before upgrading fully.
Upgrading from 2013 to SharePoint online (Office 365)
There is no upgrade path from 2013 to SharePoint online, the only real option is to use a third party tool such as Share-Gate. There is a really good Share-Gate Article which details how to migrate using their tool and the other options (which are limited!)
In SharePoint 2010 structure was important where as in 2013 it’s really important not to decide a structure for the backend (eg Site Collections) that represent your organisation structure as these can change. The navigation should be controlled by the site instead. You may find that HR want different navigation to Finance for example. SharePoint 2013 introduces search driven navigation.
There are 3 editions of SharePoint 2013;
SharePoint 2013 Comparison Chart
There are features missing from Office 365 vs On-Premise installations.
A SharePoint Farm can be a server or collection of servers and is defined by how many Configuration Databases (ConfigDB) there are as you can only have one per Farm.
Each database has a specific purpose, these are detailed in Database Purposes on Technet.