Renting is... interesting, to say the least. Anything you are likely to find online is probably over-priced or will be a major unknown. The best advice I might offer to someone coming here to live for the first time is to find something on one of the gringo expat group sites for short-term then put out all your feelers and look around for something more permanent after you arrive. Often the best deal, the most wonderful place, might just have a little hand-written sign in the window or someone in a local tienda you tells about something available by word-of-mouth.
As much of a paradise we are all convinved we have here it is a very different world. You may find it an adventure, to be certain, going through the inevitable transition but an open mind and willingness to accept a new alternate-reality is a must when coming here, and rentals might be one of those first over-whelming shocks. There are few words I could offer to prepare you for a very different world. Once you made some adjustment you may wonder why the rest of the world doesn't operate this way, especially the one you currently know.
Do not start your journey with any long-term committments, or any expectations of any kind for that matter. Any idea about what you think you may want, how you may wish to live, or what you life will be like here is bound to go through changes. Be flexible and open to change. Your first foray into this world might best be done by way of gringo connections as that is the only way to experience something that is the least bit familiar to you right now, they are the only ones that have any idea what you may expect here. Ease your way in soft and slow to start, then when you are better prepared for an alternate reality you can become more adventuresome.
Many/most homes here will have no closets. Your gringo-rented home might though. A very good chance your rental home may not
have a water heater installed, unless you know the people offering the home are from the world you currently know. There are too many unknowns and variables to caution you about. On the other hand some landlords will have everything you would expect to find, most things anyway. When we moved back into our old colonia (neighborhood) again I rented a house across the street for Tere to have her nephews that she raised to be able to live close by. Her son and his cousins (the nephews) have that home and Tere likewise has her own private get-away room there also.
That house was fully equipped and well kept and we got it for $2,000mx per month (about $165-170us). That was a good find and was there because we knew the neighborhood and knew the place may be available. It is not in the Centro but is well-connected by combi to go there very easily. It is a very nice home with three bedrooms, two baths, a dining room, and a garage. It is a nice home btu that is not to say it is not a little quirky, though normal by the standards or lack thereof here. In both bathrooms the shower area is in middle of the baño, just the shower spigot and control knobs coming out of the wall at about the midway point of the room. We have to use two shower curtains to close it off and keep water from going all over and getting everything wet, including the paper roll by the toilet. But everyone adjusts and lives with the design or perhaps lack of planning for these things.
Another thing odd is the brand new water heater. To light the pilot you do not stick-in a match or lighter but has a closure over that and a spark button to light it. It sometimes can be difficult to light after the propane tank has run dry and a new cylinder is in place, holding the pilot button while air is replaced by the new gas and you hold the button waiting for it to ignite, but it eventually does. the other odd thing is that the pilot flame is very small, which saves gas, but when the heater re-light to re-heat the water it often does so with a "boom!" out on the back patio where it is located, not always but usually does this. I don't use the shower there and have my home across the street but the nephews tell me that that Boom! is a sound they have come to expect each time about two minutes into their shower. I have inspected the water heater and find no damage done by this and maybe conclude that is just the way it is supposed to work. OK, just another unexpected thing I have come to expect here.
I'm sure someone will ask me or maybe my description of our other rented house as "fully equipped" could be mis-leading, but no, it does not have closets either. Don't expect closets to be a standard part of the house. Be thankful if yours does have them. Another thing, do not have high expectations that your kitchen will have hot water on tap. To have all the little things you may now take for granted you should expect to be paying for a very "high-end" sort of home.
Anyone looking for short-term rental that does have these things might want to contact me. I have some friends that own some apartments in the Centro of Morelia.
"There is no route out of the maze. The maze shifts as you move through it because it is alive. "
— Philip K. Dick (VALIS)