There is indeed more information needed to figure out exactly how these three variables interact. This information can be phrased in several ways, but they all boil down to something like: how much heat escapes the gas during compression?
If you're doing this compression in a totally insulated chamber, where no heat can flow between the gas and its environment, then the compression is roughly* adiabatic. Alternatively, if you're in a chamber that's not completely insulated, but you do the compression quickly enough that heat doesn't have a chance to escape, that's also a roughly adiabatic process. The point is that no heat leaves the gas during this process. In an adiabatic compression, pressure and volume are related by the following formula:
$$P_iV_i^\gamma=P_fV_f^\gamma$$
where $P_i$ and $V_i$ are the initial pressure and volume, $P_f$ and $V_f$ are the final pressure and volume, and $\gamma=C_P/C_V$ is the ratio of specific heats at constant pressure and volume for the gas in question, which is also related to the number of degrees of freedom of the gas molecules. Combining this formula with the ideal gas law should give you enough information to determine the final state. Generally, in an adiabatic process, the temperature and pressure of the gas will both increase when it's compressed.
In contrast, if you're doing this compression in a chamber that's in contact with a heat bath, and you do the compression slowly enough to let the maximum amount of heat escape the gas as it's being compressed, then the compression is roughly* isothermal. In an isothermal compression, the temperature of the gas does not increase during the compression. Applying the ideal gas law will also tell you that the pressure will increase. This can happen because the heat that would have raised the gas's temperature during compression was allowed to escape to the environment.
And, of course, if you allow some heat to escape, but not the maximum amount, then you're doing something in between an adiabatic compression and an isothermal compression. In that situation, the temperature of the gas will increase, but not as much as in the adiabatic case. The pressure of the gas will also increase, by an amount greater than in the isothermal case but less than in the adiabatic case.
*"Roughly" here refers to the fact that these are all idealized descriptions, where the minimum possible entropy is generated. If you allow the gas to leave thermodynamic equilibrium as it's being compressed (which always happens to some extent in real life), then your results may not be quite the same as those above.
