Lamp control by time of day

Hello! I have experience with high and low voltage electronic circuits. I'm not so familiar with the parts to pick which items fit together in the Arduinoverse.

Here's my situation that I'd like to solve... I have a reptile enclosure with a dual lamp fixture. The wiring of each lamp is independent inside of the hood. The lamps are basically "Y'ed" inside the lamp hood, then a single on/off switch controls both lamps. Finally, a single 6 foot power cable goes to the AC outlet for supply voltage. One light is for heat. The other is for visible light.

My simple answer would have been to use one of those lamp timers that have the day night cycle and like 50 positions around the wheel. And wire this into the hood somehow. But I can't find one of those with 2 separate circuits. Just the one timer switch would cut off heat as well as the light at night. That's not great. I can put two of those together, but now the bulk of two of these is a consideration.

In the first "version 1.0" of this, what I would like to do is power the whole thing from the one 6 foot 120V power cord. The Arduino should have a DC power adapter, and Bluetooth adapter (for changing on/off schedules when needed) or other easy interface, and two high voltage relays to control the on/off cycles of each lamp based on a time of day schedule. What items would I be considering? I'd favor small and compact if it's an option. The lamps are each 120V 2A. Is this even feasible?

In the future I might enjoy having a temperature sensor and shut off the heat lamp if room temp is warm enough. (No need to waste electricity) Also it could warn me when the heat lamp is on, but the bulb is burned out (no heat is sensed). I might also look into a photosensor for the lamp as well. But both of these are version 2.0 of this device. Right now I'd like to get 1.0 up and running!

It's easy. Get yourself a dual relay board, and an RTC.