Disclaimer: I am not a professional HVAC person. I have fixed the ones on my own residence and bought a couple of complete systems before.
The furnace, inside air conditioner coil (A-coil), and outside air conditioner should all have stickers on them somewhere that give a model and serial number. The stickers are usually on the outside of the A-coil and outside unit, and on the inside of the furnace - take the front panels off like you were going to change the filter to see it. Usually there is a number or two in the serial number that tells you the year the unit was built - Google the brand, like, "Lennox serial numbers" or "Trane serial numbers" or similar to see how to read it. Taking a photo of the sticker for later reference sometimes works, but zoom into the photo right after you take it and make sure you can read everything... the furnace sticker is sometimes in a dark corner and the outside sticker sometimes gets faded in the sun. This doesn't immediately affect anything, but if the units turn out to be older than you think, maybe that will help you make a decision.
The model number will also usually tell you the capacity of the furnace (in BTU/hr) and the air conditioner (sometimes tons, sometimes BTU/hr). You may have to Google to learn how to read the model number - for instance, a 60,000 BTU furnace might have "060" or "06" in the model number at a certain spot. For that size house in New Orleans, you might have a 60,000 or 80,000 BTU/hr furnace. I am less sure about the air conditioner - 3 tons or 36,000 BTU/hr maybe? Knowing the capacity will help an HVAC company give you a ballpark estimate for the equipment. If you're taking photos, also stand back a bit from the furnace and take a photo of the whole thing, including the ducts going in and out. This will tell an HVAC company if the furnace flows up, down, sideways, or what. They have furnaces for all of those setups but they need to know which one to bid.
The air conditioner stickers will also tell you what refrigerant (Freon) they use. R-22 was the old standard that is going away, R-410A is probably the current standard. This basically tells you if your previous service tech was telling tales or not.
It is possible to replace just the air conditioner if you want (the outside part and the coil above the furnace). The HVAC salescritter will probably try to talk you into replacing the whole thing, but you don't have to.
The houses I have lived in and owned, in Missouri and Oklahoma, have had Lennox, Carrier, and Trane systems in them, and they have all seemed to work OK. The Carrier A/C on one of those houses had the same Freon in it from the day it was installed in about 1969 until the day it was replaced in about 1995. It still worked, but the owner wanted to replace the 30-year-old Lennox furnace and get a more efficient A/C at the same time. The Freon system is made up of copper tubes that are soldered together into a sealed system when the A/C is installed - if the installer puts the right amount of Freon in at the beginning, it stays there for the life of the system. It's not like a car radiator, where some of the coolant can evaporate out over time.
The Trane system I had installed at my residence was really slick - the installer put a specified amount of Freon in it by hand, plugged an electric valve on his Freon bottle into the outside A/C unit, and pushed a "fill yourself up" button on the outside A/C unit. The A/C opened the valve and ran itself until it had as much Freon as it wanted, closed the valve, and blinked an LED to tell the installer to unplug his bottle. (The tech was also watching it with the pressure gauges they usually use, and he commented that it closed the valve right at the pressure he was expecting.)
The maintenance things I have had to do with them include: replacing the igniter on a gas furnace (about every 6-10 years, $30 for the part), replacing the run capacitor on the outside unit (once in about 15 years, about $30 for the part), replacing a fuse (once in about 10 years, about $5 for the fuse). For a tech to do these things, it would have been probably $100 for the truck roll each time, and $50, $50, and $10 for the parts, respectively.
Every year or two, I also cut the power to the outside A/C unit and hose out the coils with a garden hose. They get dirt, cottonwood seeds, and other junk in them, which makes them less efficient. I let it dry for a few hours and then turn the power back on.
I also change the filters. The $1 fiberglass filters are good for a month and the $3 pleated ones are good for 2 or 3 months. (Try to find a local store that sells a box of 10 or 12 filters, if you can - cheaper that way.) If the tenant has cats or dogs, use the pleated ones and change them every 1 or 2 months.
@Andy Gostisha 's advice about checking for rebates and deals is right on the money. It looks like Entergy is both the electric and the gas company, but if not, make sure to check with the gas company too if you have a gas furnace. Note that sometimes for the rebates, you have to have someone check the old system first - either testing its performance or looking at the model numbers - in order to get the rebate.
I hope this helps!