Free parameters in the Standard Model

Yes, wikipedia has a table which lists the 19 free parameters that need to be tuned by experiments. These include, as you already said, the masses of the elementary particles including the Higgs Boson, and some other notable ones are:

• CKM Mixing angles and CP-violation phase.
• Gauge coupling of the three symmetries (U(1), SU(2), SU(3)).
• Higgs VEV

In the standard model, technically neutrinos are assumed to be massless. According to special relativity they would travel at the speed of light, so they would be timeless and could not oscillate. Hence, the 3 neutrino masses and 4 mixing angles are technically not parameters of the standard model. However, in simple extensions to the standard model, these parameters are free as well.

It is important to note that the 19 parameters in Wikipedia's table are not the only possible choice to describe the 19 degrees of freedom. i.e there are multiple choices of parameters that span this space.