Generally speaking there are 2 types of wire: Stranded and Solid. Most of the time you want stranded, it's more flexible and is far less likely to break off at the solder point. Many people call wire that you solder in small electronics projects "hook-up wire", but it isn't a standard term.
Next is the size of the wire, known as the "gauge" or abbreviated as "AWG". For small electronic projects 20 to 30 gauge wire is what you'll probably use. The smaller the number is the LARGER the wire is. For low voltage data wires 28 and 30 gauge works perfectly fine. That rainbow ribbon wire that you bought is usually stranded 28 or 30 gauge wire so it will work for the majority of your project. However for the wires that go from the charging port to the battery you'll want to use larger wire, probably 20, 22, or 24 gauge. The reason that you want larger wires for the charger port and battery is because smaller wires have more resistance and since a lot of electrical current goes to the battery while it's charging you want as little resistance as possible.
