I am trying to build a system to control lights remotely, but I am a student and am operating on a tight budget. Is there any way I can have just one Arduino along with a receiver and transmitter (I am using the 433Mhz RF Wireless Transmitter Module and Receiver Kit 5V DC 433MHZ) to do a simple on/off command for lights? I don't want to unnecessarily spend money on another Arduino (especially since I am not going to be transmitting complicated data). If this were possible, what are the other things I would need and what would the code look like?
It depends on your hardware. You are not transmitting complicated data, but you need to transmit some data. Just wiring a switch to a standard RF transmitter is not enough. Though there are RF modules, which support exactly this (by having corresponding electronic components on the board). If you haven't already bought the modules, you could buy these.
If you already have bought the RF modules, you can buy a standard 433MHz key fob. They are normally rather cheap. Then you can record the key fobs signal with your receiver and write code to check for this signal in the received data.
If you plan to keep on building with Arduino, you might wanna buy directly a bunch of Arduinos (also for future projects). You can get Arduino clones directly from china for very cheap (a Nano clone for example costs normally between 1 and 2$ each in packs of maybe 10).