How fast does heat travel through conduction, convection, and radiation? Does heat have a specific speed or does the speed depend on the type of material it's going through? Like how long would it take for the heat from the sun to reach Earth? For example if the sun just went out for a second, how long would it take for us to feel the cold? Would we see it go out first or feel the cold first?
http://coolcosmos.ipac.caltech.edu/cosmic_classroom/light_lessons/thermal/transfer.html
 A: 
How fast does heat travel through conduction, convection, and radiation?

You are looking for the rate of heat transfer $\dot q$ (joules per second).
Conduction
Fourier's law states the amount of heat conducted a certain distance/depth into a material per second, when there is a temperature difference:
$$\dot q=A \kappa \frac{dT}{dx}$$


*

*$A$ is contact area,

*$\kappa$ coefficient of thermal conduction (material specific),

*$T$ temperature and 

*$x$ depth (distance the heat has "travelled").


The term $\frac{dT}{dx}$ is the so-called temperature gradiant (often written as $\nabla T$). For a symmetrical object, this is simplified to $\frac{\Delta T}{\Delta x}$.
Convection
Newton's law of cooling states the amount of heat transfered between the surface of a submerged object and the fluid every second:
$$\dot q=Ah(T_{s}-T_{\infty})$$


*

*$A$ is contact area (surface area if submerged),

*$T_s$ surface temperature of object,

*$T_\infty$ fluid temperature far away from (not affected by) the object and

*$h$ (often $h_c$) the convective heat transfer coefficient (material, fluid and process specific).  


Convection is not simple. Convection can be forced (as by pumping) or natural (as by buoyancy), and the relative velocity $v$ of an object moving through a fluid, viscosity $\mu$ etc. play big roles. The equation appears simple at first sight, but includes the parameter $h$, which takes into account the process-specific details and may be very complicatedly tied upon other factors. This source shows some rough examples of values of $h$ in different situations.
In specific engineering cases, you either estimate (or numerically model) $h$ or apply a correlation  of $h$ with other parameters that someone have found, if such exists for that specific case. Or you experiment and measure your way through it.
Radiation
The Stefan-Boltzmann law states that any surface with a temperature above $0\;\mathrm K$ radiates heat constantly:
$$\dot q=\varepsilon A\sigma T^4$$


*

*$A$ is exposed (uncovered) surface area,

*$\sigma$ Stefan's constant $\sigma=5.67\times 10^{-8}\;\mathrm{\frac{W}{m^2K^4}}$,

*$T$ temperature and

*$\varepsilon$ emissivity of the object (material and surface specific).


An emissivity of $\varepsilon=1$ gives an ideally radiating body, a so-called blackbody.

Note: Literature might often omit the $A$ in the three laws above. The laws are then stated as heat fluxes instead of heat rates. When dividing through with $A$ the left-hand-side becomes $\frac{\dot q}A$, called heat flux (sometimes given the symbol $\Phi$ or $\Phi_q$ or $\Phi_h$), which is simply heat transfered per square meter every second. 

Does heat have a specific speed or does the speed depend on the type of material it's going through? 

This is not an either-or case. Heat moves at a specific rate and that depends on the materials involved. See the few parameters in the description above, which depend on material.

From what I understand heat travels through conduction by 2 objects at different temperatures.

Heat only flows if there is a temperature difference, yes. The fact that you have two object is irrelevant - same is the case within one object. A temperature difference across two points will cause heat conduction, if they are in physical contact.
A: As I understand it, while heat itself does move at a constant and determinable rate (known formula), the actual 'effective rate' in any specific situation is very much complicated/effected by the physical properties of the materials involved and the specifics of the surrounding environment. This too is mathematically determinable (known formula) for any specific and unique situation, but all of those many impacting variables of materials and environment can be difficult to define precisely for the formula.
