That's an interesting question. It depends on whether Mach's Conjecture is true or not. Mach's Conjecture is, basically, that inertia is the consequence of gravitation effects of all the rest of the matter in the cosmos.
If the MC is true, then the effect of the teleportation to an empty cosmos would be to turn off inertia which would have profound and difficult-to-estimate effects on everything. It would almost certainly cause matter to become unstable so that the Earth dissolved into some sort of primordial goo of subatomic particles. Or something else very fast (less than nanoseconds) and very final and not at all helpful. (No one as far as I can has tried to calculate the details as the effort would be both non-trivial and non-useful.
If inertia is not affected, things are a lot easier to estimate.
First, it would get dark.
After that, the next step would be the loss of GPS and other satellite-based services after a couple of hours. Most satellites are solar powered and have batteries only big enough to deal with the 45-minute eclipses they normally see. Sun goes away; satellites switch to battery; batteries run down; satellites go dark.
The tidal bulge in the oceans and solid Earth would no longer be supported and you'd certainly see increased tectonic activity as things settled down, and you'd also see something like a day or so of world-wide, smallish tidal waves as the oceans relaxed to their non-tidal states. This would mainly be in the few-to-many hours range.
Then it would get cold.