Ok, a few questions.
1. How do I save in this game?
The game is constantly saving. If you quit out of the game you will start right bakc where you were.
2. I heard you lose blood echoes if you die, and permanently if you die twice in a row. I have 709 blood echoes, should I just buy one of the armour pieces available that are 500 each, or can I just find them in the world as a drop item?
A lot of stuff you can find as drops. Most all of it, in fact. As long as you are looking around and being thorough you will find some new gear.
3. How do I upgrade my characters stats?
After you get your first point of insight, usually by encountering the first boss, the doll in the hunter's dream will awaken and you level through her.