Thanks a lot for all the help so far but I am building a cell incubator for growing mammalian cell cultures, not a chicken incubator.

Oh, so eggs aren't mammalian cells now?!

Going back to your heat-producing device that you linked on omega.com, that's a really overpriced heater. We've already talked about using light bulbs, but if you have a concern about light in your incubator then the next option would be to use a power resistor.
Aluminum clad power resistors would work well if you can attach them to metal walls/etc. inside the incubator or you could probably attach them to a CPU heatsink/fan to circulate hot air through the incubator. The resistance needed is calculated by R = V
2/W where R is the resistance, in ohms, W is the wattage rating of the resistor, and V is the voltage you intend to run through it. You should choose a slightly higher resistance to stay safely under the wattage rating of the resistor. The lower cost of these resistors (~$3 for 10W) also means you can spread many of them throughout the incubator for more even heating.
SouthernAtHeart linked to a popular temp/humidity sensor and
here's a good tutorial. If the accuracy of these devices is too low for your liking you'll need to provide more specs for us.
Back to controlling the on/off of the heat, an electromechanical relay is fine if it will only be switched on/off a couple times a minute. Excessive switching will wear out the relay or drive you nuts with a clicking noise, so if you need to switch more frequently then you need to go to a more expensive
solid state relay. These are non-mechanical devices, and when switching AC current they can be switched on/off -- assuming a 60Hz line frequency -- 120 times per second. Note that this linked SSR is not designed for DC current; those get a little more expensive.
For a reliable, interactive display I would suggest the
Phi Panel. It plugs right into an Arduino, and if you get stuck and need support then you can count on
Liudr to help you out with it. There are also Chinese knock-offs that might work fine but it'll probably ruin your chance at a friendship with Liudr

.
Finally, to glue everything together programming wise, the
PID library will give the most accurate results. The PID library would look at your current temperature reading, your desired temperature, and then figures out the the best timing for switching your heaters on and off to keep a stable temperature. It will provide better results than typical "if temperature < setpoint turn heater on" type of code.