when or how can I start building new things besides the defaul three towers and the two barricades?
you can start building new things by clicking at the tiles then select if you want to build the wall/trap or the buildings , you can only build things that you have unlocked.Build barricade on the path and towers beside the path.
To unlock new buildings , you need to level up your Inventor’s Workshop first and then you can also level up the construction plans
Theres also the castle gate , upgrading this will give access to more tower and barricade + path limit.Theres a max number of 17 towers , 13 walls/traps and 29 paths for max level castle gate.You can design the path according to your own preference and space limit.Some of paths will be block by ruins , it will cost lots of gems to remove them so it is better to build path on what tiles you have first.
I moved the topic to a more appropriate section btw