The very first tool you should buy for your Fiat 124 isn't a wrench. The first tool isn't a ratchet or a screwdriver or even a pair of pliers. The first thing you should buy to work on your car with is a shop manual. There are several popular ones, written to high standards, that have been published over the years, so the first thing to do is get one that covers the model year car you have. It is no use getting one that ends with 1978 if you have a 1982 Spider convertible. Most of the contents about the electrical system and the fuel delivery will be very different.
The big advantage to having the manual right from the start is the reference material it contains. You can trace electrical faults on the wiring diagrams. You can find components like the fuel pump or the starter motor based on the photos and diagrams they all have. A shop manual is an excellent way to orient yourself as to the general layout of the car and also the specific components.
Then there are the procedures included. Every one of the top manuals has chapters that cover servicing the brakes and the cooling and heating system. There will be step by step instructions on how to replace suspension parts and troubleshoot the wiring. There are also going to be some very good guidelines about regular maintenance. This can be the most important aspect of owning a Fiat 124 or 2000 Spider.
Fiats from the late 1960s to the early 1980s respond to care. They are not a car that you can just drive and put gas in, topping up the oil when the dashboard light comes on. That is the recipe for walking home. Adhering to some sort of service intervals is the only way you are going to get more than a few miles out of your classic.
There are several very good Owners Workshop Manuals still available for these cars, even though the youngest is over 20 years old. You just have to know where to look. At yard sales and car parts swap meets, on eBay or Craigslist or even listed used on Amazon. They will range in condition and price from missing covers and covered in greasy smudges for a buck or two, to brand new, still shrink-wrapped versions for $30 or more. I recommend something in between.
Which one to get? There are several brands out there, from Haynes or Chiltons to Autobooks and Clymer. Each one is a little different, but they all share some common features: Lots of diagrams and black & white photos, tables of torque specifications, repair and rebuilding procedures and often wiring diagrams and troubleshooting tips.
Before you buy the first socket, combination wrench or timing light, get a shop manual and at least browse through it.