Adding wires to existing board. Not sure where to solder them
Soldering would be the best bet, as it would provide the most reliable connection for your specific application. Use a temperature-controlled soldering station at about 300 degrees C to prevent delamination of copper. Also use ample flux to prevent shorts. Use strong, tacky flux and as much as possible.

It's nothing bad if you make a short. It's always fixable as long as you don't delaminate the copper. Apply less heat in short intervals. Be gentle like you are writing with a pencil you won't want to break the pencil tip. Also you can use lead or lo temp solder. You can also operate on the test points which you have circled in the picture.

Use a low guage wire if you are using stranded wire, twist and tin first.

After you are done, fix the wires with hot glue or tape so that the solder pads are not subject to mechanical stress.