There are multiple ways to do this.
The most-common one in the US (for complex modeling) is Blender with an MMD import-export plugin. Free. You have to deal with file format changes and such, though. More for advanced modelers.
A lot of Japanese models are made with a program called Metasequoia, which is paid software. Simple interface and easy to use, but costs more if you want better import-export functions.
A bunch of the MMD folks also use PMX Editor, which doesn't have the modeling tools of the others, but is MMD-native (and also free).
I really suggest going to https://learnmmd.com/ for their tutorials and links. You can also find a lot of models and helpful tutorials on https://www.deviantart.com/
I think MMM is superior to MMD in almost every way, including animation. For instance, you can scrub through the animation timeline.
Here's the link to MMM:
It's Aegisub. http://www.aegisub.org/
Also look up for: Kara Effector (used for editing the fonts and so on)
Most people use effects already created, a simple google search for "aegisub karaoke effects" will show you many of them.