Welcome!
You can certainly brew on the stove, I did it for a few batches no problem. It's slow, but it works.
Since you're renting, like me, I'll assume you have little storage. I would highly recommend the BIAB (brew in a bag) method. This way, you really only need one pot, a heat source, and a bag.
For a bag, I have great experience with these guys out of NS:
http://www.mashedin.beer/collections/custom-brew-in-a-bag-biab For a heat source, if you're using the stove you're probably going to be limited to 5 gallon batches. I used a standard shitty apartment stove to boil 6gal for a 5gal batch before. It works, but it's slow. A great way to get your feet wet anyway. If you have a balcony or outside space a cheap $50 burner from Canadian Tire works well.
For a pot, there are lots of good options out there. The best deals seem to be at OBK for a pot with a thermometer and ball valve. Based on the Stove's limitations, a 9gal pot will be enough for most 5gal batches, but might be tight. I started with an 8gal pot and it worked good, but I wanted some more room so I upgraded to the 15gal.
At this point, you really will want a wort chiller as an ice bath will take FOREVER and it's best for your beer if you chill it quickly. The longer the beer spends at near boiling temps, the more bitter it will become and you'll lose hop aroma. Also, a quick chill will create a "cold break" which will help drop out some proteins and lead to clearer beer.
Ideally, you would have the ability to control the fermentation temps consistently. Some people use their basement, some people use a "swamp cooler" (google it) and some people have built a fermentation chamber out of a fridge or freezer with a temperature override. Temp control is one of the most important aspects of GREAT beer.
So- really all you need at this point is the pot, the bag and the chiller. A burner is optional. If you decide to go with a mash-tun set up in the future, then all of this equipment would be usable, aside from the brew bag.
Edit-> You don't need a mill right away either. You can buy pre-milled grain from Noble Grape. I did this for the first few batches. You can always lean on people here to help you out for a few batches too.