The most popular form of towing is all 4 wheels down, but there are advantages and disadvantages to dinghy (4 wheels down) and dolly (2 wheels on a dolly) towing. You will most likely chose which method depending on what the capabilities of your towed vehicle are (or you end up buying another vehicle).
Generally, advantages of 4 wheel down towing are simplicity, quicker to attach-detach, and no dollys to store.
Disadvantages are they are more involved to setup, and require modification of the vehicle (towbar brackets and other items), and usually require some kind of braking system (most states require brakes on the towed vehicle if over 3,000lbs), tail lights, battery charging, and so on. As well, they can put more wear and tear on the transmission so it may not have as long as a life, or at least will require an accelerated servicing.
Advantages of dolly towing (2 wheels on a dolly) are that more vehicles can be towed - almost all 2WD vehicles can, but some low clearance vehicles might not. But 4WD vehicles and most rear-wheel drive vehicles cannot. And dolly towing is more simple, you just drive on the dolly attach the straps and safety chains, and you are done. As well, the dolly usually has a braking system, and it's own tail lights (but you might have to supplement this depending on your state law). No towed vehicle modification is required.
Disadvantages include a bit more effort to connect and disconnect the vehicle, and you need somewhere to store the dolly. Also, the dolly adds around 600lbs of towed weight and probably 200lbs or so tongue weight. And handling may be more difficult as there is more un-sprung weight on the dolly.
There are more advantages/disadvantages to each method, and that can differ depending on the specifics of your towed vehicle, but this is pretty much a generalization of the major characteristics.
Unfortunately, most automatic transmission vehicles cannot be towed 4 wheels down. Only a few of these vehicles can be towed, and lately, it seems Ford has the most towable models with auto transmissions. You risk severely damaging your transmission should you tow a vehicle without having that capability.
In addition to the owner's manuals, Ford publishes a comprehensive Towing Guide which has a section for RV towing, detailing all of the vehicles that can be towed. It is interesting that depending on the transmission/drivetrain, some model Fords can be towed and some cannot.
Most, if not all manual transmission vehicles can be towed, along with many 4x4s that have a transaxle neutral - such as the Jeep Wrangler.
And some vehicles can be towed for some model years, and some cannot. For example, one of the most popular tow vehicles; the Honda CR-V cannot be towed for the 2015 model year as they went to a CVT transmission.
Other vehicles, such as the Jeep Cherokee (new Cherokee - not the Grand Cherokee), can be towed or not be towed, depending on the transmission. Only the optional high/low range Cherokee 4x4 transmission can be 4 wheel towed. The 2 wheel transmission can only be dolly towed, and the lower-cost standard 4x4 transmission cannot be towed at all.
The best way to determine if you can tow your vehicle is to check your owner's manual, or download the owner's manual for the year vehicle you are looking at. Most manuals have a section called "Recreational Towing" that will let you know if you can tow it or not behind a motorhome, and what restrictions there are - if any, as well as any specific procedures (disconnecting the battery, etc).
To start your search, the best method, I think - is to download or obtain Motorhome Magazine's "Dinghy Towing Guide". It is published every year, and is a good starting point as it lists almost of all of the vehicles that can be towed. But follow up with your find with examining the owner's manual for that vehicle.
Also, a few other vehicles, such as the Honda Oddysey may be able to be towed, even though the owner's manual does not recommend it. Here is where talking to, or observing other RV'ers (what they are towing) is of benefit.
is a good source as well. They list the different transmissions, which can be towed and which cannot, and recommend whether or not a transmission pump can be used.
And you can also sometimes put in a transmission pump to convert a vehicle to towing capability, but you are going to need to invest a couple of thousand dollars for the pump and it's installation.
A lot more older vehicles with automatic transmissions can be towed. For example, most Saturn automatics can be towed. So it might behoove you to find a used vehicle that you can tow.
I tow a 2002 Pontiac Grand Am, which has the same automatic transmission many Saturn vehicles use; hence it is one of the few automatic transmission vehicles that can be towed. Generally most every model year Grand Am, and it's G6 replacement can be towed, and some Pontiac Sunfires can, but no other Pontiacs can. The other GM siblings to the Grand Am probably can be towed as well.
The Grand Am tows well, and is around 3,000lbs, so it is one of the lighter vehicles that can be towed. I was able to buy it with 39k miles on it last summer from a relative, and since it is fully depreciated, I did not cringe as much by drilling holes into the frame and grille work.
Also pay attention to the max towing capacity of your motorhome, as you will not want to exceed the towing limits.