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If people could travel in floating vehicles, fly with the help of a device and travel across the world in airships, would they still require roads designed for cars?
For instance, if you had something that looked like a car, but hovered, would it still require an asphalt road? Or if it flies high in the air like in Harry Potter, would we need some form of highway?
At the crux of your question, "Would we require roads?", probably not.
The answer could depend on a few factors:
Where there land-based travel forms before the floating technology? Or was the floating technology developed first? If the former, then its likely roads would have been needed at one time, but eventually abandoned ( assuming floating travel was easier/not more expensive ).
How much are people walking? Heavy foot traffic can sometimes necessitate, or even create, small 'roads'.
Would you consider dedicated air pathways as roads? If so, it likely that the shortest routes between cities would be filled with traffic, much like a ground road.
Would we require roads? Maybe. Depends on your world.
Also, consider breakdowns and the Napa lines (which some say are used for aerial guidance, even if they aren't true roads). Roads may be present as guidance and may be present in case of an emergency landing.
A certain number of ground vehicles are always going to be required: for example agricultural equipment such as combine harvesters, tractors to plow the fields. Also it may not be cost effective to transport freight by air. Heavy loads, or large awkward loads might still travel by land.
So roads would be less important, but still exist. Major highways might become one or two lane only.
Flying has its downsides:
I can probably think of more, however the point is that whilst the technology could be available it wouldn't necessarily be better.
Think of the first couple of points: These would be expensive vehicles to run with the constant maintenance (a lot of personal cars at the moment don't get that sort of treatment and break down regularly - a definite no no in mid air). Only the rich would probably run flying cars. However, lets assume everyone does and continue our thought experiment.
Flying may well be banned within cities to make sure no one hits a building and it probably wouldn't be faster than normal cars unless we only use the tops of buildings.
On long distances between cities they might be used if power isn't an issue since you aren't restricted so much by terrain and can choose a straight route, don't need to break or turn - they could even be auto-piloted along theses routes to avoid collisions.
Essentially yes, we would need roads within cities since the maneuverability offered by friction cannot be rivaled by flying vehicles. Unless you're only using the tops of buildings (and they're all the same height) you will want your transport to be able to weave between - much better in cars.
The answer to your question is No, there will be no need to design roads for cars, given access to flying "personal" vehicles by everyone.
Your question is getting taken ... for what ever the reason, to be about, exclusively, "commercial" BULK flight, NOT personal flying vehicles.
There are currently several companies on the verge of or actually offering "personal FLYING drones".
Here is a link to review "https://www.engadget.com/2016/01/06/184-delivery-drone-for-people/"
NASA and the FAA are or should be testing to implement an autonomous controller for these types of vehicles but then again they have been developing this for the last 17 year or so and one can be quite right to ask Were is it, why is it taking so long.
Here is a link to review "https://www.nasa.gov/feature/ames/first-steps-toward-drone-traffic-management"
If you look at wilderness areas often you'll find that travel is mostly done by air, sometimes water.
The problem with air is weight. Moving a handful of people or a cargo of small items can easily be managed with ease in a small aircraft. If you need to move a bulldozer, industrial equipment, heavy freight, or mass transit then you need very big planes, at which point transport cost becomes a factor.
The C-5 Galaxy is one of the biggest airplanes in the world, and is routinely used to move tanks. It has a lift capacity of 122 tons.
It's also not the best way to transport things. For its voracious consumption of fuel and its maintenance and reliability issues the Galaxy's aircrews have nicknamed it FRED, for: F*ing Ridiculous Environmental Disaster.
A train on the other hand can pull a load in the tens of thousands of tons range. Not as fast though. (Determining how much a train can pull is apparently a complicated process.)
Container ships can haul loads in the hundreds of thousands of tons. Also not as fast.
Airships such as blimps are an option. Hydrogen has a net lift of 71lbs per 1000 cubic feet of gas. Helium has a net lift of 68lbs per 1000 cubic feet of gas.
The LZ-126 / ZR-3 Los Angeles had a useful lift capacity of 31 tons using helium, 50 tons using hydrogen.
The Hindenburg had a lift capacity of 10 tons IF hydrogen was used for the lifting gas. If helium was used instead the lift would be -17 tons, meaning it would stay on the ground.
TL;DR: It depends on what you need to move, and how you want the "magic" to work. If it's just people and light freight then no problem. Once you start moving very big things then air is less attractive unless magic says it is.
Scenario 1: Air transport is not only cheap, but small. Because it's cheap to use people have spread out. This results in massive numbers of commute vehicles moving about. This scenario requires aerial lanes to constrain traffic. So you have virtual roads. I don't think this is practical in urban areas with out a totally automated piloting system.
Scenario 2: Air transport is cheap, but requires dedicated fields for launch and landing. In our world we call them airports. No between city roads needed. You still have in city roads. Note that the landing operatings for a zeppelin were messy, and each one requires a ship length circle around it's mooring mast. Zep parking lots are going to be big.
Scenario 3: Air transport can take the form of a hover car -- something that floats a small distance above the surface. You still need lanes, but they don't have to be paved.
