All campaigns eventually come to an end. Hopefully it was because of the natural end being reached instead of being left unfinished. Regardless, we have to make decisions about what we will do next and these decisions can be extremely important for the success of the new campaign. It's also not a very simple subject. As usual, I hope my thoughts on the matter will be of some help.
Rolling Into the Next One
One of the big things that need to be decide is whether a new campaign will be started right away or a break will be taken. When I say right away, I don't mean necessarily that things will continue on the old schedule. The beginning of a campaign has a way of often being more complex than a normal session because many details about the world, characters, player characters and the general story going forward needs to be thought up. However, you can still have the first session of your new campaign scheduled in 2 or 3 weeks. I typically like to do it this way because I find it is much easier to keep things rolling if the people involved know when the next session is. Trying to gather people back together after a break of unknown length can be its own challenge. In some cases, this is very hard to avoid. If it will be a month or so away it can be very difficult for people to predict if they will be free. However, if they already liked playing in your games and want more game time they will typically give it their best shot.
Switching the Dungeon Master
Rotating the roles can be a great way to mix things up. There are some people who like to play and some who like to run the game. If that's how the group runs and they are happy, that's great. However, I've also seen groups where people enjoy being on both sides of a game. It can also be helpful for a Dungeon Master to get some time to let their ideas stew. Being able to have a campaign going and still working on their next campaign can be an amazing blessing for a Dungeon Master. One thing that I will warn about, however, is some unintentional leaching. You might think of things that would make a great addition to the campaign you are playing in but would be out of place in your own. You also want to be wary of having your next campaign being too similar to the current one as well.
New Campaign Length
One of the decisions that tend to be made is about the length of a new campaign. Many people want to just keep the game going until interest is lost or they run out of things to do. Some, however, prefer to have an ending in mind and work towards it. A short campaign may just span levels 1-5. The nice thing about shorter campaigns is that they can be extended and built on if needed. It also means that a suitable and manageable big bad can be chosen for the planned range. The risk is that things may seem a bit disjointed if no thought towards the next possible extension is made before the end. I find many people make these kinds of decisions even if they don't realize it. These choices can be seen from elements such as world building, enemy choices, magic items presented and sometimes even player advice (when it comes to character building, some builds work better than others at certain level ranges).
The end of a campaign can be a good time to switch groups if things weren't going well. I also find that it's easier to handle when a player leaves at the end of a campaign compared to leaving in the middle of one. That way, it gives time to find other players if needed instead of scrambling in the middle. There is also the barrier of getting the new players caught up and integrated so they don't feel left out, which is avoided when a new campaign is being made.
Taking A Break
Personally, I view this option as a last resort. I find it much harder to find the time later instead of when I already have time scheduled. It can also be a bit hard to jump back in due to some of the reasons I mentioned earlier. Campaigns can run longer or shorter than expected but they typically run a few months at least from my experience. However, if in a slump it can be a great way to get rid of fatigue and get excited about the game again. I've seen some people for who this works great. Personally though, I prefer playing anyway and the fatigue goes away on its own when new, exciting situations are encountered.