I am using components like these:
How should I be connecting this to the breadboard. I am now inserting wires and pulling them to the sides but whenever I touch the breadboard, contacts disconnect and that is making the debugging process so much harder.
I am thinking of soldering little wires, but can I solder something this small (should be like 2.5 mms between inputs) for the first time and be successful?
The best way is to get a pin header and solder it into place.
The pin headers look something like this:
The standard distance is 2.54 mm, which should be same as the distance between holes on the breadboard and on the PCBs. These pin headers are usually easy to break into smaller sections, so you can buy a bigger one if the exact size is unavailable.
The exact type you want would be 2.54 mm pitch, single row straight male pin header.
To add to AndrejaKo's answer: this image shows the indents where you can break the header to your desired length. Longer ones, like 30 pins, are often cheaper per pin. This one is gold plated, but tinned versions will do for soldering. Make sure you get the right pitch, 2.54mm (they also exist in 2mm pitch).
If you want to remove the PCB afterwards to use in another project you want sockets as well.
These are not breakable, so you have to buy the right size. These are usually used with gold plated headers.
