This may be a dumb question but does matter always contain mass? And alternatively, does mass indicate matter? I mainly ask in the case of subatomic particles and black holes. Also in the sense of E=mc^2, anything with energy should have mass? 

Fundamental fermions gain the property of mass through the Higgs mechanism.
That means electron, quark, and neutrino families.

Any composite particles they make up also have the property of mass, but it is not a strictly linear addition.
As mass and energy can be equivalent properties, altering the configuration of a system can alter the energy, or the mass, of the system.
This is usually seen in terms of binding energies.
As an example, the fermions that constitute the proton/neutron are combinations of 3 up/down quarks, where the mass of the fermions makes up just 2 % of the total mass; the other 98 % is binding energy.

The W and Z bosons (which mediate the weak interaction) are massive, so having mass is not an indicator of being matter.

Matter is an indicator of having mass

On 11/7/2020 at 11:05 AM, Xechs said:

This may be a dumb question but does matter always contain mass? And alternatively, does mass indicate matter? I mainly ask in the case of subatomic particles and black holes. Also in the sense of E=mc^2, anything with energy should have mass? 

Can but not should. Photons for example do not have mass, but can contribute to the mass of a system that they're in.