Imagine that you call a cab. It arrives and you step in. You tell the driver your destination. He drives there, deals with the traffic, maybe you have a little chat. You arrive, the driver tells you the fare and you pay it, and you step out at your destination.
That is teleportation in action. Your magical teleportation differs from this in following, non-essential, ways. You call it with magic. It doesn't look like a cab. It is much faster. And it can presumably pass thru intervening space without interacting with it noticeably. Although most settings do have some methods for blocking teleportation, so some interaction with intervening space can be assumed.
So basically your question can be answered by answering the question: "How do you notice that the (invisible) cab has arrived." The most obvious ways are the sight and sound of the cab, in your case these would be special effects you can freely choose and can be ignored here.
With a cab there would be also exchange of air, but the fact that teleportation skips interaction with intervening space causes an issue here as it requires a change to some non-interacting phase at the origin and back to normal at the destination. This requires either that the destination is linked to the origin before departure or that teleportation works like some sort of accelerated astral travel where interaction needed to "shift back" is possible in transit.
Usually these two forms of travel considered separate and only the former is considerde teleportation. That means that two points are connected in advance of the teleportation and the arrival or departure of the teleporter would be preceded by a a gust of wind caused by the pressure difference possibly causing scents or even light objects being carried to the point with lower pressure. In that way it would be more like stepping thru a doorway than using a cab.
You can ignore the motion vector issues, any plausible form of teleportation has to match the frames of reference. If it doesn't it isn't teleportation, it is a cannon blasting the target with sub-atomic particles thru obstacles. Pretty useful but not really relevant. And using metal for ammunition is cheaper than forcing people to step in such a device.
As for limits, you can model these thru the cab analogy. Reasonably knowing the teleportation spell would be like owning a car. It would require maintenance and refueling between uses. It would have maximum carrying capacity in both the size and mass of what it can carry. And it would have a maximum usable range based on the amount of fuel in the tank. So you could do a second jump right after a short hop, but would have to split long distances to shorter hops with refueling stops. It would also have a maximum speed that would drop with mass carried, but it is probably not worth modelling that, unless the plot requires carrying somethin near the maximum mass limit.