Who will win the technology race: VacMagLev, or hypersonic planes?

The correct answer is neither because neither are particularly sensible means of transportation.

Barring that, train good plane bad

VacMegLev would be the correct answer. It's just the best way to move things, energetically speaking, and by conceptual simplicity. It might be hypersonic planes, though, because all you need is the plane and a blatant disregard for pollution or expense. I really hope not.

Suborbital deserves a mention as a sort of in-between option. It coasts through vacuum most of the way, so between antipodes it might actually beat conventional passenger service, for theoretical cost. I should do some napkin math sometime.

What about rocket slingshots and space tethers? Couldn't they work the same as a suborbital, but cheaper?
- I'd split the napkin calculations into "passenger ICBM" and a more efficient scenario where the rocket just gets you to altitude, and the rest happens by skyhooks that recycle all their energy.
  
  Ah shit, I'm doing this, aren't I? Stay tuned for the edit.
  
  Edit:
  
  Alright, to get vertically up, it's around 2 km/s delta v. It would make sense to launch from a jet in the stratosphere, Virgin Galactic-style, so we'll neglect drag, and assume this is all lost on re-entry. (Maybe you could use the heat for manufacturing somehow? Another time)
  
  To get to an antipode suborbitally on your own, it looks like it's the same as for low-earth orbit, which is kind of obvious in hindsight. So, 8 km/s. Distances in between are going to be in between, and we'll measure by fuel used.
  
  Now we pull out the rocket equation, and assume an (effective) exhaust velocity of 2 km/s, for a really pessimistic case, or 20 km/s for an optimistic scramjet-type case. Let's assume each passenger counts for 200kg of dry mass, including luggage and the craft itself.
  
  (e^2km/s ^/ ^2km/s-1) * 200kg ~= 444kg propellant.
  
  (e^8km/s ^/ ^2km/s-1) * 200kg ~= 10,700kg propellant.
  
  (e^2km/s ^/ ^20km/s-1) * 200kg ~= 21kg propellant.
  
  (e^8km/s ^/ ^20km/s-1) * 200kg ~= 98kg propellant.
  
  Meanwhile, for a conventional airline, we'll assume 2L/100km per passenger to make room for modest future improvements in efficiency. For an antipodal trip, that's 400L * the density of jet fuel, so roughly 320kg.
  
  Scramjet cases all win out, and the bad rocket case is in the same ballpark with skyhooks (but takes only a couple hours and you get to be weightless in space). The trick, of course, is that none of these run on pure jet fuel. Conventional rockets will probably will run on some hydrocarbon and some LOX, and a scramjet is very chemistry sensitive and will need something special like LH2. LOX looks like it comes at around 1USD/kg in bulk, while A1 is going for 0.81USD/kg at the moment, so that's not a huge issue. Hydrogen is easy enough to make, but LH2 is god-awful to handle to the point where even SpaceX doesn't want to bother.
  
  There's also still a lot of unknowns about the expense of maintaining hypersonic airbreathing craft - it might well dwarf fuel costs in even the most advanced, refined case. I don't know how easy a few hundred kg per passenger is to achieve with conventional rockets going into near-orbital trajectories. There's also the issue of upper-atmosphere pollution, which could hurt the ozone layer.
  
  I think the conventional + skyhooks case is most interesting scenario, looking at the issues and advantages. Virgin Galactic is taking a b-line for profitability and mass access, and there's startups working on the kevlar skyhook. Who knows, maybe this will be a thing starting in the 2040s?
- Done.

Honestly I just want to see us slowing down and gaining more free time and flexibility, as our ancestors had. What's the problem with taking a month or two to sail or to fly in a solar powered airship? All this haste is plain nonsense to me and I don't think it's healthy for humans.

Going around the earth in 3 hours would require you to travel at 11 times the speed of sound, and that is without including the time it would take to accelerate or slow down.

The concorde flew at a maximum speed of twice the speed of sound. It would take the concorde 18h30m to fly around the world if it had enough fuel to do it.

Supersonic travel has some major issues. It takes a huge amount of energy to go that fast. Concorde could only cross the atlantic ocean, because it didn't have enough fuel to cross the pacific. The other issue is sonic booms, which means you can't fly supersonic over populated areas, like land.

Maglevs have the same issue as all other high-speed overland transport, it requires expensive infrastructure to be built the entire route. The faster you want to go, the flatter, smoother and more expensive the track will be to build.

Supersonic air is more plausible as it only requires a faster airplane. With wealth inequality, there are rich people who can afford their own supersonic plane, but an infrastructure project to build a global maglev network is far more expensive than that.

Maglevs have the same issue as all other high-speed overland transport, it requires expensive infrastructure to be built the entire route. The faster you want to go, the flatter, smoother and more expensive the track will be to build.

Yes, and going over or under the ocean is a bit of a pipe dream right now. It should be possible in principle, but nobody has the details worked out, let alone the economics.
Also, supersonic air travel already exists. It's not a technical hurdle anymore and the Concorde was even profitable in her later years.

There's a lot of research addressing the very real problems you pointed out and it seems plausible that they'll at least be mitigated to a degree in the near future.

Hyperloop is a (vacuum) pipe dream. It sounds super cool, but the more you think about it, the less realistic it gets.
- I said hypersonic not supersonic
- I've always heard that seismic activity makes hyperloop-style transit effectively impossible.*
  
  † Impossible given the constraints of current materials science. From what I've read (which may be garbage, but it seemed well researched), making a vacuum chamber that's hundreds of thousands of miles long on top of a big wiggly molten goo ball isn't something we can even see a way to realistically achieve right now.

A rich asshole can buy a hypersonic plane for suborbital jaunts.

Vacuum maglev pipes require cooperation from the people whose land you’re going to take via eminent domain.

eminent domain doesn’t require cooperation
- It only requires a police force.
Vacuum maglev pipes require cooperation from the people whose land you’re going to take via eminent domain.

Do you think that kills the idea? We have two major cities with 300km of mostly farmland in between where I live, so I'm pretty sure it would be no problem here - if investors would fund it.
- the people that own the farmland might object
You could build them along existing tracks (not that it's a good idea in the first place)
- Higher speed trains need much larger turn radius so existing RoW is not suitable. They’re running into the same issue trying to build high speed rail into the SF Bay Area.

Why does one have to win? Don't both have their own strengths and weaknesses?

I would love to take a train across the Atlantic or island hop over the Pacific but those are some serious engineering problems.

Cities don't pick trams, light rail, or bus networks. They have a mix of all of them.

Hypersonic passenger flight has some serious weaknesses. I'm not sure it will ever make sense when there's billions of people who need to be served and a finite planetary energy budget.

I could see billionaires doing it, and in theory they could even manage it in a green way (but don't hold your breath). Most Lemmings don't like the world working like that, though, and it's not really in the spirit of comparing it to municipal transport.