First off, your bike has a Sachs 504 engine. Most likely the "a" variant. Sachs engines are very simple, and it's a great bike to understand the inner workings of a 2t..
The Sachs 505 is the same engine but the pedal shaft is integral to the engine, while yours is separate...the 505 was available with the "d" variant engine, a different cylinder and longer crank stroke gave the bike much more power. More than enough with good reliability...the good news is I believe that your engine cases are compatible with most 505 parts. Soo you could run a d top end I think.
Sadly, the clutches on Sachs engines are the weak link. Unlike other makes like Puch, motobecane....the clutch cannot handle the power of increased displacement...so if you want to up the bike to 70cc, your in for a world of hurt.
Also, your bike was actually designed for a Puch e50 engine. Look at your engine mounts, if there is an adapter bracket you can remove it and run any Puch moped engine. Much simpler clutch design and can hold alot more power if you ever wanted to tune the bike.
Phew, anyway read the general articles in the wiki here. And check out the Sachs page for engine specifics...
You need a basic metric socket set, some wrenches. A nut driver handle for your sockets, it's nice to use a socket like a screwdriver when you need to....... A multimeter is good to have too
Feel free to hit the pm button below my post to send a private message if you need help with wrenching. I have alotta Sachs experience (edited)