In order to clarify entirely what I mean, I am only concerned with transfer orbits such as Hohmann, bi-elliptical, etc.
It's none of those. Both are for transferring from one circular orbit to another, and with both orbits on the same orbital plane. Neither Earth's nor Mars's orbit is circular. Mars's orbit is notably non-circular, with an eccentricity of 0.0934. Mars's orbit is inclined with respect to Earth's orbit by 1.85 degrees. This is more than enough to make a Hohmann transfer a useless fiction. (Aside: Even if the orbits were circular and on the same plane, a bi-elliptical transfer wouldn't make sense for transfers from Earth to Mars. The orbits are too close to one another.)
What is done instead is to solve Lambert's problem, over and over and over. Given a central mass, a pair of distinct points in space $\vec r_1$ and $\vec r_2$ relative to that central mass, and a pair of points in time $t_1$ and $t_2>t_1$, Lambert's problem involves solving for the conic section that starts at $vec r_1$ at time $t_1$ and intersects $\vec r_2$ at time $t_2$.
Except for diametrically opposed points and radially aligned points, there are two such solutions to Lambert's problem. One, "the short way" (or sometimes, a Type 1 transfer), involves a change in true anomaly that is less than 180 degrees. The other solution, "the long way" (or sometimes, a Type 2 transfer), involves a change in true anomaly that is more than 180 degrees. Note: Some of these solutions might involve going faster than the speed of light. That's not a problem in Newtonian mechanics. It is a problem with regard to delta-V. Regarding those diametrically opposed points and radially aligned points: These are problematic with regard to Lambert's problem. The solutions are singular. The general approach is to ignore those points.
Suppose the point $\vec r_1$ represents a point on some initial orbit at time $t_1$, and point $\vec r_2$ represents a point on some target orbit at time $t_2$. With this, one can calculate the delta V needed at time $t_1$ to transfer from the initial orbit to one of those Lambert trajectories and the delta V needed at time $t_2$ to transfer from that Lambert trajectory to the target orbit. This is the delta-V cost for that particular trajectory. One of the two solutions will have a higher cost (typically much higher) than the other. We'll discard the high cost solution.
Now do this over and over and over again, but with different departure times $t_1$ and arrival times $t_2$. Generate a contour plot with departure dates and on one axis and arrival dates on the other, and eventually you'll come up with something like this, from
"Porkchop" is the First Menu Item on a Trip to Mars:
This is a porkchop plot. The blue contour lines show the energy (here in terms of C3) needed to transfer from Earth to Mars in 2005. (Note: Since we're interested in getting from Earth to Mars (or back), it makes more sense to use C3 rather than delta V.) Note that the transfer orbits are markedly non-Hohmann. A Hohmann transfer would take about 275 days. The optimal transfers instead take about 200 days (short way) and 400 days (long way).
I am doing a school project that requires me to calculate the math behind an orbital transfer to mars for a human mission.
You didn't mention what level of school this project is for. If the above is over your head, you can ignore all that and use the fiction of Hohmann transfers. If it's not over your head, this is exactly what is done to calculate transfer orbits to Mars at JPL.