Exporting a Database schema and triggers
To find information about database object ( I am using triggers as example here )
use TOAD 
choose Triggers
Right Click on Triggers
Create Script 


That will give you source for all triggers

Repeat this for all TABLES, INDEXES, SEQUENCES etc. After reverse engineering a few - you will get hang of it.


Connections between tables


I believe you mean "relations between tables". If you schema is small say a few tables, then you can browse thru Parent and Child option of each table and document them.

For large number of tables, you will need to use TOAD to create Data Model Diagram; or maybe google for "Data Modeling Tools" and find one that fits your needs and budget.

You will imperatively need to refer to data model diagram many times in your new roles - so do it patiently and correctly.