To be able to instantly travel to another location is a staple in fantasy and the Teleport spell and other similar spells have been a staple in every edition of D&D. It has (like so many other things) being treated mainly from a game-mechanical perspective with a bit of mishaps and errors added. What if we instead looked at teleport (and similar spells) from a setting perspective, how do they work, wat limitations, dangers and quirks could we add and derive when it comes to the ways and means of magical transportation. The spells of interest usually falls under the Conjuration School, which doesn't gives us a good list of which spells that will most likely be affected by the ideas to come.
There are real world ideas for teleportation using what we know of quantum entanglement and other way out there schemes. For our purposes we only need to make it cool, reasonably easy to use and first and foremost make it fit a magical setting. I'm going to propose that there are more than one way to accomplish this, each with its own set of advantages and disadvantages.
Astral
Making use of the Astral Plane makes it possible to escape the limitations of the primal world, either in mind only which is fairly easy, or bring your body as well which requires much more of an effort (Teleport is a 7th level spell after all). The advantages are that you can reach a large part of the multiverse using this method and are not limited to places you can see or know well.
The drawbacks are that you need to either know the destination well, or have some magical means of targeting it (more on this later). Your travel can be detected and interfered with by those in the planes you need to traverse in order to get to your destination.
Ethereal
Travel using the Ethereal and ignore things like terrain and even the planet itself. It easier to master and also harder to detect at a distance for anyone not in the Ethereal Plane using magical means.
Less powerful than the Astral method, you can only reach across the prime (using the border Ethereal) or deep into the Ethereal itself. Another drawback are that traces of your journey can linger for quite sometime and can be seen and traced by those that can visit and properly sense the Ethereal.
Fey
The Feywild offers interesting and in ways powerful forms of transportation, like Tree Stride for example. It is very hard to detect and fairly easy to master, but comes with its own set of limitations. The main one is that it requires things that are somewhat connected to the Feywild, which are usually living in nature, and it requires their cooperation or at least acceptance for it to work, and it requires the creatures in both ends to be in on it. Creatures usually store memories, and can have motives and allegiances, sometime they can shift which can open up for interesting problems.
Shadow
Using the Shadowdark (Plane of Shadow) for your quick travel needs can be very convenient, shadows are found in lots of places, (usually) just a medium without an agenda. Hard to trace and even harder to detect from a distance, it is a very stealthy means of transport. It comes with risks, shadows are by their very nature fickle and can disappear as quickly as they appear. They are almost limited in what parts of the multiverse they are strong enough to be used as a mode of transport.
Elemental
Using the elemental planes to travel can be an tempting alternative to going there by normal means. Large bodies of a single element often have openings to its elemental plane, and portals to the elements can be opened using magic. A key advantage of using elemental travel is that it is very stealthy, away from the prime, astral and outer planes where most of able observers reside. Most of the elements are murky or solid (except air) which means senses are usually very limited in range.
A major problem with elemental travel is the very nature of the elements, they can be harmful, restrict breathing as so forth. Another major problem is the elements still adhere to the laws of the prime meaning travel speeds are as fast as they are on the prime. Elemental travel might be the sneakiest way around but it takes time, can be very risky and doesn't take you to large parts of the multiverse.
To know where you want to go and how to get there can be way more difficult depending on the nature of your destination, and its surroundings. The locations in the prime are easy to get to if you can see them or are intimately familiar with them. Intimately familiar means you have been there for a length of time, and crucially, you know how it relates to the place you are going from. So for a teleport to function well you need to both know where you are, and how you got here from your destination. This can be tricky if you have used magical means to get to where you are or demi planes and other forms of plans disturbances are at play.
Gates and Portals
Gates and portals are a way to overcome some of the problems reaching your destination by creating a place that have way to aid magic travels, in some cases just provide a higher degree of certainty for those who cast a teleport spell and other a permanent connection open for all to use for travel.
Gates and Portals can be very complex and require keys, special triggers like spells or only work at special times or conditions. Some of them are set to a specific destination, others can shift due to circumstances.
Forks
Planar Forks are a kind of "teleporting" key that can be set to guide you to a certain place. They are magical in nature and requires the creator to either have been to the location in question or its true name (which can be for a place as well as a creature). True names are very hard to find and understand which makes Forks very valuable and hard to create and come by.
They are magical meaning they can be detected, and they don't make the use of teleportation more stealthy.
Beacons
Like a Fork but it works at the destination and sends out its magic to guide teleporters to help them steer right. They have a range and work in a medium of some sort, like audio, smell or light in the prime. Beacons can also be way more special, working in the Ethereal, Astral, Shadow or Fey, sending pure magic, good or evil for miles around, sometime even across whole planes. This is up the the DM to create some flavorful beacons, like the Eye of Mordor for example, that has more than a single purpose.
The general rule is that the more useful a beacon is the more conspicuous it is. Beacons can be detected as far away as they are useful as a guide, often even farther away.
What happens when you miss your destination, something or someone messes with you on the way, or the destination itself is not stable enough to be there when you arrive.
Missed Destination
Magic travel is almost always inherently risky, and you might not end up where you intended with mild to catastrophic consequences. How likely this is depends on several factors like knowledge of destination and how to get there, the way you travel, magic or other forms of interference. Some forms of magic travel are riskier than others like shadow that are very shifting by nature. The current Teleport spell provides a good base table for missed destination.
Detection
The chance, or rather risk of, being detected while magically traveling doesn't have to mean anything, but it can be ruinous depending on the destination and/or purpose of the journey. The chance, range, means of detection and information obtained depends of where you travel and hoe the detection is attempted.
Here you can be both logical and a bit fantastic. Determine what kind of senses works where and then add in a bit of uncertainty, like protection spells that hides your identity but not prescence for example.
Detection can work both ways, the traveler might be able to detect attempts to actively scry for example.
Interference and Interception
This is a both the most interesting part of this discussion, but also the one that can ruin the fun of using magic transport in your game. It needs to be applied with reason and logic (fantasy logic that is). Magic forms of chase rules might work in some instances. Make sure that interception is a process that requires detection of enough information to be able to act, then some for of action, magical, sending a creature to intercept etc. Then the possibility for the Traveling party to detect the interference and maybe even avoid it or counter it. This is where you might run in to an astral dreadnaught trying to kill you, a shadow or fey monster ambushing travelers, or fiendish creatures on the lower planes always on the lookout for good smelling souls.
Sometimes it is enough to be able to slightly interfere with the magical travel and send the travelers off course instead of intercepting them. This might be a way to lessen the change for the traveler to know that someone detected their journey and influenced it.
Deception
Risky situations, detection and a hostile environment gives rise for the need to be able to deceit and fool your opponent. There are many ways this can be done, sending a decoy to se if it goes through, using teleport without sending anything and making use of illusions and enchantments to fool senses and minds of others can be used in creative ways to further your goals and provide a safer, more stealthy travel.
This was some short introductory thoughts on magical travel. I'm currently working on adjusting the existing spells to fit this way of looking at the topic, adding new spells and features to deal with the specifics. A closer look at both sensing and magic dispel will be needed, but I like a bit more depth to this. A general rule I have is that if something happens often the rules need to be simple, but if it happens rarely the rules can be more complex, and magical travel is not that common. It is also something that I see lacking in flavor so deserves more depth.