Does it matter what solder you use?

Absolutely!

Bernzomatic silver bearing rosin core is plumbing solder, not electronic solder. You need to get either 63/37 of 60/40 (Tin/Lead). Both have their advantages and disadvantages. There are also different kinds of fluxes within these solders and wire diameters.

You can do a lot of research (here's a good summary). Keep in mind that soldering is as much an art form as it is a science. People will argue up and down on which solder and flux to use like their lives depended on it. Ultimately, everyone develops their skills and finds their own preferences.

However, you can't go wrong with a 63/37 solder with No Clean (NC) flux at a diameter of 0.020". Look for Kester, AIM or MG Chemicals for their solder line up. Order whatever is most convenient.