Recently, I had a really cool idea. A new campaign in a homebrew world that I'm making revolving around some kind of wild magic "incursions". The first encounter takes place in a tavern, of course, and the players don't know each other yet. They order drinks, and they find themselves seats. That's when the screaming happens. Sounds of running, screaming, general panic are happening outside. A few moments later, the wall explodes inwards and gibbering mouthers come crashing in, eating everything and everyone they can. Inevitably, the PC's will fight them, and win. They step outside and see strange rifts that looks like tears in the air that horrible monsters are stepping out of. However, the city's military has gotten it under control relatively fast, with their fighters killing the monsters and their mages closing the rifts. I love this idea and think it sounds awesome. However there's one huge problem. I dont know where to go next, and I want this campaign to go for a long while. Any ides?

Well, if I were playing in this campaign, I’d assume the idea was for the party to figure out what’s causing the problem, and stop it. 
So, maybe decide on the cause, and how it can be stopped, and let the players figure it out. Along the way, they can resolve issues from their backstories, and solve other side problems as they come up. 
Fyi, this basically sounds like the plot of Dragon Age:Inquisition and/or Elder scrolls: Oblivion (and probably a dozen other places). You might look there for inspiration. Don’t make one character the main character, of course, but you can probably steal some ideas from those places.

Something I forgot to say, I do have an ending. The big bad, the whole reason that these incursion are happening, is a child. An insanely powerful sorcerer that is being controlled by a lich. The lich has a cult that believes this child is the 'chosen one' and will help them to bring the world into their control. The lich manipulates this child with magic, charming him and changing his own appearance to look like the kid's dead father. Because the little kid is so young, I want him to come in during the epic final battle to add a moral dilemma of whether or not to kill him. The problem is I don't know what to put in the middle without them just finding out immediately who it is and going and killing them.

Have you seen the old Ghostbusters movies? Also a nice place to borrow from for that kind of campaign.

Sounds pretty cool! I personally would maybe make it so the town council is asking for a response team and the PCs volunteer, but there are other ways you could probably do it.

Quote from carsorino >>

Something I forgot to say, I do have an ending. The big bad, the whole reason that these incursion are happening, is a child. An insanely powerful sorcerer that is being controlled by a lich. The lich has a cult that believes this child is the 'chosen one' and will help them to bring the world into their control. The lich manipulates this child with magic, charming him and changing his own appearance to look like the kid's dead father. Because the little kid is so young, I want him to come in during the epic final battle to add a moral dilemma of whether or not to kill him. The problem is I don't know what to put in the middle without them just finding out immediately who it is and going and killing them.

 You have them work their way up the chain. They find some minor cultist, who points them toward a slightly higher ranking cultist, and so on. 
And I’d throw in a side plot. The first one coming to mind is a local ruler trying to exploit the chaos for personal gain. While the neighboring kingdom is dealing with all these tears in reality, he senses weakness and launches a war to claim some territory on the border. I’d make it completely unrelated, but make it seem like it’s not. It only looks like the guy has stirred this up as a distraction, that can lead the PCs in a wrong direction for a bit. Or, he could be allied with the lich, that can work, too.

And maybe there’s a Druid circle working in opposition, since these things are very unnatural, so the PCs can try to ally themselves with the druids. But, of course, they’ll need to prove themselves first.
And, as I said earlier, pepper in some character-related side quests, and just some other 1-off side quests, and you’re good to go.

Think in terms of bad guys. Who is causing this and why? What is their plan, and how do these incursions fit into it? Are these a side effect, or is invasion the goal? What clue remains behind after the first encounter for the players to find, and where would that lead? A few questions I always like to ask when homebrewing adventures, hopefully they help!

Sounds like you need to figure out why magic has gotten weird. To my mind, the obvious implication is that something in the Far Realm is preparing a way for itself to enter the material plane. Gibbering mouthers might be a first wave of attack or just a symptom of whatever it's doing.

Ideally, you want to have an ending in mind. That's not to say you'll get there in play, but it is how you work out what happens before the ending. So for example, I would make the next section of the game about getting to a library in order to establish some lore about what's happening in the world. But I'd need some enemies, so, as other posters have pointed out, I might introduce a cult that has some connection to this inimical entity that's harming magic. That also might give me an allied group. If you've got a god or goddess of magic they'll be interested in what's happening, and their church might support the PC's actions.

I'd continue planning until I knew whether the ending I wanted was, "evil thing from beyond the stars breaks free" or "players foil summoning ritual thus preserving the realms" or "magic is forever altered in this setting" or what have you. My planning wouldn't be finished until I worked out that ending, because knowing the ending would help me know what kind of cult opposes the PCs and what kind of allies might help them. Once I knew the ending, I could choose an interim antagonist--some leader of the cult--and I could give them motivations that align with the ending I had in mind.

Good luck!

Quote from carsorino >>

Recently, I had a really cool idea. A new campaign in a homebrew world that I'm making revolving around some kind of wild magic "incursions". The first encounter takes place in a tavern, of course, and the players don't know each other yet. They order drinks, and they find themselves seats. That's when the screaming happens. Sounds of running, screaming, general panic are happening outside. A few moments later, the wall explodes inwards and gibbering mouthers come crashing in, eating everything and everyone they can. Inevitably, the PC's will fight them, and win. They step outside and see strange rifts that looks like tears in the air that horrible monsters are stepping out of. However, the city's military has gotten it under control relatively fast, with their fighters killing the monsters and their mages closing the rifts. I love this idea and think it sounds awesome. However there's one huge problem. I dont know where to go next, and I want this campaign to go for a long while. Any ides?

 Given that the city military has it under control you'd probably want the city to commission the party.  Some one likely will need to ask the party to help otherwise the military driving the forces back may give the impression that the party doesn't need to intervene.  You could also lead them around to it  the long way by giving them odd jobs that lead up to it but that will be comparatively difficult and there is really no point in broadcasting a big event and then slowly working up to it. Once you have an opening like this the main plot isn't a secret and odd jobs seem like distractions.

Very cool ideas and a good start to a campaign but I'd suggest a few refinements and perhaps a bit less planning at this point.

1) You have a picture for the final battle already in your head. The problem with that is you don't know what the players or you will do between now and then. You don't know how your world will evolve and grow. You don't know what other factions you might put in. You might even decide that the lich controlled child isn't the end. Perhaps something nastier has seen these portals as an opening to a new world to conquer, completely separate from the lich and the sorcerer savant who accidentally opened them in the first place. Having at least two POSSIBLE BBEG and an evolving story line lets you adapt to the decisions the players make.

2) I think your opening scene is too advanced along the plot line. If the town guard has guards and mages ready to respond within minutes fighting an incursion and closing rifts then this has been going on for a long time already. The characters should not be surprised since they would have already heard of these incursions if they are so common that by the time the adventurers come out of the tavern the guard is already getting it under control.

My suggestion would be to back up this plot line a bit. Have the initial event of the campaign be the first incursion in this town. Perhaps have some rumors that some village 50 miles away was attacked by strange creatures that the unbelievable stories claimed came out of thin air. Since a child is creating  the rifts, perhaps by accident or as a side effect of some other training or perhaps having a nightmare, they could close when the child is interrupted or wakes up or for some other reason completely unrelated to the characters and party.

Wherever the rift appears, a few random creatures from this other plane, enter and attack folks. Don't make them too tough to start with and have the party assist a few town guards fighting the creatures that appear. Involve the characters in the first scenario.

After that, the local crown or government, who have no idea what is going on, decides to offer a reward or hire adventurers to try to figure out what is going on. They have their own people working on it as well and can perhaps provide additional information or advice to any other groups investigating the problem. Emphasize to the characters/players that they aren't the only group working on this.

In addition to the main plot line, throw in some side quests and other activities that may or may not be related to the main objective.

The problem with having mages and guards fight off and close the rifts in the first scene is "What are the characters supposed to do?" .. there are all these mages and guards who have the problem in hand and are likely already trying to fix it - what can a bunch of low level adventurers hope to accomplish if the government already has rift closing mages looking into it? You want to create a story beat that will allow the characters to get involved in a way that makes sense.