This method may help you add shape of the roads.
Your satellite navigation system will work, but up to some limit. Accuracy of consumer-grade units is limited (~7 meters) unless you make complex and expense processing of them using tools like RTKLIB, industrial-grade antennas and specific satnav chips.
Ensure that your GPS unit has the 'Snap to road' or 'Lock on road' option switched OFF, or you will be recording a direct copy of the map loaded into the GPS unit, which may well be a copyright infringement.
When gathering data with your GPS (see Hardware Guide), additional data needs to be recorded in order to add details to the map: street names, amenities and anything else you wish to map. You can record this additional data with pen and paper, a voice recorder, or a camera (cellphone camera or dashcam).
Using JOSM, you can import photos, and it will place them along your GPS route by using the time stamp. This assumes your camera records the ‘second’ at which each photo was taken. For people using cameras that don't record the 'second', taking a photo every time you pause allows you to easily link collections of dots to points at which you took photos, and link them accordingly. Or, you can Synchronize GPS and camera.