A big ditto on the solid core wire! You could also “tin” stranded wire with a bit of solder in a pinch if you don’t have the right gauge solid core. 22 AWG wire is perfect. You can pick up “kits” with pre-stripped sections for prototyping. But if you’re planning on doing more than one project don’t waste your money. Just buy a spool and strip it yourself. I work with network cable so I use this. It works quick and smooth. If you cut the wire to the exact length you need you can keep it from “rat-nesting”. A needle nosed pliers will help bend the wire around other componants on the board. Any wire coming off the the board can be bundled together and then taped down to the breadboard so it can be moved. If you do end up soldering, save yourself some headache and go with this solder. It has a lower melting temperature so it is quicker and less heat on your components.
Also, I like putting resistors across the devide of the breadboard. It seems like it gives you more room to work with, and it just looks nicer. That and some other good tips can be found here.
It looks like you are already using Fritzing, in case anyone else is wondering how to make a schematic like that it is free software, but they do ask for (optional) donations when you download. I was amazed when I discovered it. It lets you record your breadboard design visually and lets you take that over to a circuit diagram. Even just using the breadboard mode makes it really easy to save a working layout before breaking it apart. It has almost every Arduino already in the database, you will have to add the Spark Core graphic / schematic to use it for your project. Luckily someone created it!
Fritzing Download Page
GitHub Repo for the Spark Core schematic file
To add the spark core schematic right click on the “Parts” panel in Fritzing to the right and choose import. The file from that github repo that you need is in /lib/SPARK_CORE_v10.fzpz