Some, I think. My brain wants to understand it as if a particle is made up of the same stuff as an antiparticle, but with opposite charge, but that seems a little too simple. Is that closer to correct, or am I even farther off?
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The elementary particles are not made of anything, as they are elementary. For those, what I said above applies.
For composite states, it is sufficient to replace any constituent by an anti-guy. For instance, a particle by the associated antiparticle and an antiparticle by the associated particle. I think this agrees with your brain :)