In Sleeping Dogs, unlike the Grand Theft Auto series, you’re not exactly given a gun straight away to start ripping up the city with. You have to search for them and find them.
One of the easiest places to find weapons though are on police officers.
If you find a police car, you can search it and you’ll possibly find a shotgun inside.
If you’d rather have a pistol though, try to find a police officer walking around the city. You can either knock them out and take their gun from the ground, or just disarm them to take it.