Basically, you need a mixer with mic preamps and aux sends and you need effects. These days for a live show a digital mixer is certainly the way to go.
There are two broad categories of what we could call "digital mixers". There are dedicated hardware digital mixers, and then there are DAWs and software mixers that you install on a computer that is connected to one or more audio interfaces.
With a hardware digital mixer, you generally get everything (mixing, signal routing, and effects) all in one box. Most of the time, what you get in a hardware mixer is what you got, so whatever effects it comes with is all it can do. The advantages of hardware digital mixers is that they are cost effective, reliable, low latency, and easier to set up and use than software based solutions. Often you can change the settings on hardware digital mixers from anywhere using a wifi connected tablet. The Behringer X25 line of digital mixers is an extremely popular example of these products.
With software, you first need a fast computer and a fast audio interface in order to apply the effects you want in real time without excessive latency. The audio interface provides all the inputs and outputs to the system, the computer provides the processing power, and the software uses the computer and interface to route the audio and apply effects. The advantages of a software based system is that you can completely customize each aspect of the system, and you can add third party effects, so pretty much any effect you can imagine, you can use (if you pay for it). The downsides are that these systems are more complicated, more expensive, and can be prone to more kinds of failures than dedicated mixer systems. In terms of software, Ableton Live is very popular for live sound processing (hence the name), and Apple Mainstage is also used by many big shows. Mainstage requires an Apple Mac computer to run, and can be controlled remotely with an iPad using free Apple Remote software. Ableton Live can be controlled remotely with an iPad, but you have to buy a customizable MIDI/OSC app like Lemur, set up the Lemur interface, and then manually map the Lemur controls to Live. This is a much more complicated setup but is also very flexible. Another good control option for Live is using Ableton's own Push 2 hardware interface, but that connects with a USB cable and is not wireless. Also, Push 2 is not really designed for mixing and effects, it's focused on triggering beats and loops and samples.
On the software side, there is one hybrid option, but it is likely the most expensive route. There are interfaces that have dedicated outboard DSP, the Universal Audio UAD system is probably the most famous one of these. The advantage of these systems is they give you very low latency, high reliability, and all the flexibility of choosing what effects you want to use, but the interfaces and effects are generally quite expensive.