The type of solder is not very important. Lead-free solder can be harder to work with and requires a higher temp. Leaded solder is not environmentally friendly, but is much easier to work with. To start with in soldering, you want to remove the old solder and apply flux before re-tinning and soldering the part.
This video series is quite old, but is really the best tutorial series I have ever found for soldering. I know it is a bit boring, but it is well laid out. They don’t go into lead-free solder, as it was a new concept when these videos were made. Lead-free has a different luster to it and a bit higher temps required to melt, so a ‘Good’ joint with lead free will not be as shiny as with leaded solder and this is normal.
PACE - Basic Soldering Lesson 1 - "Solder & Flux"
My personal favorite solder:
Kester 24-6337-0010 44 Rosin Core Solder 63/37 .020