How many suns does the milky way galaxy have?

Somewhere between 200 and 400 billion, depending upon how you define "sun." If you define it as a star, then see above. If you omit neutron stars and various dwarf stars or proto suns (stars which have not yet begun nuclear fusion), there would be less.

If you define as sun as any star with one or more planets in orbit, we only know of 623+ so far, though again the number would likely be in the billions.