You could turn invisibility on and off as you like and there would be no time limit. Your clothes would turn invisible too, and you could decide whether the items you are holding would be visible or not.
There would be no limits on how many times or where you could teleport. The items you hold while teleporting would be teleported too. You would also have the ability to know if the place would be safe to teleport to, so you wouldn’t teleport and get impaled by an icicle or teleport inside a wall and get your insides filled with concrete or something.
Personally, I don’t know which one would I pick. Invisibility would be awesome for pranks and stuff, but teleportation probably would be more useful for everyday life.
I always understood teleportation as a accelerate-decelerate sequence to reach the end point, as in not A to B but A to B to C, where B is the midpoint where you have enough momentum to naturally stop at your destination.
A lot of depictions of teleportation seem to imply that you instantaneously move between two distinct points in space without having moved through the intervening matter - if that’s the case you have to deal with a whole lot of complexity to ensure that you are moving at an appropriate speed at your destination - it depends a lot on your reference point, but imagine teleporting from the equator to the north pole without accounting for the difference in velocity of the ground due to the earth being a sphere. If you were standing still on the equator but preserved your momentum through the jump, you’d be moving at 1600km/h when you arrived. Air friction and the sonic boom alone would mess you up, let alone if you collided with something.
You are right, you could solve it by having the teleportation move you through some sort of hyperspace and making the jump almost instantaneous so you have time to accelerate and decelerate, but even then you’d need to hand wave away where the energy goes. Imagine the same set up, you’ve accelerated through hyperspace towards the pole, then decelerated back down to end up going 1600km/h slower than you started. For an average person this is about 8 MJ of excess kinetic energy that has to go somewhere