Disclaimer: I am not a professional HVAC person. I've bought new furnaces and A/Cs for houses before. :)
Find out what size the furnace and A/C are. The furnace will be rated in BTU/hr - probably something between 60,000 and 130,000 or so. The A/C will either be rated in tons or BTU/hr - something between 1.5 to 4 tons, or 18,000 to 48,000 BTU/hr. There will also be an efficiency rating, which will be something between 10 and 14 SEER for the A/C, and 80% to 95% for the furnace. These don't have anything to do with the sizing, but they do affect the price, and they may affect the rebates you can get from the local utilities for upgrading. The home inspector may have put the model numbers on their report; you can determine the size from those, and with a little Googling, you can probably come up with the SEER and furnace efficiency.
Call a few local HVAC companies and see if they'll give you an estimate over the phone, for that size furnace and A/C, with equal or better efficiency to what you have now - some will quote you over the phone and some won't. Or, Google for ads - some of them will advertise things like "80,000 BTU/hr furnace and 3 ton AC installed for $X". Add a few hundred bucks to $X, because they leave some things out of that price, but it will at least give you an idea.
Many HVAC companies will offer a financing option. Often, it isn't real great - it's like a store credit card with high interest - but it might help spread out the cost a little. You might ask your bank how long you have to own the duplex before you can get an equity line of credit on it - you might be able to finance the systems through the HVAC company, pay that note for 6 to 12 months, and then pay off the HVAC company's 18% loan with 6% money from your equity line.
Right now, every HVAC company is busy fixing and swapping air conditioners. By October, they'll be busy fixing and swapping furnaces. There is a lull in their business sometime around September, and if you can possibly schedule a replacement then, they will sometimes give you a discount of one or two hundred dollars a system, just to keep their techs busy. They might also cut you a little bit of a deal for doing both of them at once, but I'm not as sure on this one, because I've never done it.
Also check with the local electric and gas company - sometimes they offer rebates for replacing old systems, or moving up to a higher-efficiency system. Sometimes a tech has to test your old system and then test the new system for you to get the rebate; sometimes you just have to fill in a form that lists the model numbers of the old and new equipment. The utility should tell you how it works. The HVAC companies know about these rebates, and have adjusted their pricing accordingly, but it can still save you a little money.
If it were me, and I didn't have the money to do two furnaces and two A/Cs, I'd do the two furnaces. If a furnace screws up bad enough, it can kill somebody. If the A/C quits in a way that is uneconomical to repair, a couple of hundred dollars at the hardware store or general store gets you some window units that will serve until you can replace the central air.
If you just want to replace the furnace, the HVAC people will make noise about "the system will be unmatched", mostly because they want to sell you an A/C too, and a little bit because installing a new furnace with an old A/C can make the installation slightly more complicated. (Mostly, they might have to build a little bit of duct between the new furnace and the old A/C coil, which they don't have to do if they are doing a new furnace and a new A/C coil.) Try not to pay too much attention to that noise - it's perfectly possible to make a new furnace work with an old A/C coil. If one HVAC company doesn't want to do it, ask another one.
If you do just the furnace, ask if they can have a tech work over the A/C while they are there for the install. Mostly this would include cleaning out the outside unit with a hose, cleaning the inside coil before they install the new furnace, checking the refrigerant (Freon) level, and (if they're feeling really spiffy) checking the amp draw on the compressor. Doing this stuff will help the A/C last as long as it can, and tell you a little more about what condition it is in.
It will be different in Detroit, but as some kind of reference: In the Kansas City area, with existing ductwork, I could probably get two complete, nice systems - about a 2.5 ton, 13 SEER A/C, and 100,000 BTU/hr, 90%+ furnace - installed in an average duplex for maybe $14,000 to $15,000 all in. If I used cheaper systems I could probably get it down to around $10,000, plus or minus.
