I heard that antiquarks are just like antimatter. Does that mean that they cancel quarks out? If so, does that mean that there are more quarks than antiquarks?
It is indeed true, that antiquarks are antimatter and annihilate with quarks. As knzhou pointed out in his comment, there are more baryons (3 quarks) than antibaryons (3 antiquarks).
There is also another kind of hadronic (made of quarks) matter, mesons, consisting of one quark and one antiquark each. Those, however, are not stable.
As Chappo pointed out in his comment, quarks don't exist in isolation. Thus, we cannot observe these processes with typical scattering experiments. However, from the decay products of mesons, we can reconstruct the decay process using conservation laws.
For example, $\pi^0$ (a superposition of $u\bar u$ and $d\bar d$) can decay into two photons, which can be interpreted as the quark-antiquark pair annihilating.
It is also possible for quarks and antiquarks of different flavour to interact in a similar process. Due to conservation laws, however, these processes involve the weak interaction, so they happen at a much lower rate. Instead of photons, which mediate the electromagnetic force, gauge bosons of the weak interaction are produced in this case.