i am beginner and i would like to know what equipment do i need to connect Ableton Push, Akai APC40, guitar + pedals, vocal processor and mic to play and record live? Do i need a preamp? Thanks.
It looks like the missing piece in your case is an audio interface. That's a device that has audio inputs and outputs and also some kind of computer connection, like a USB, Firewire, or Thunderbolt interface. You should look at interfaces that have at least two analog inputs and two analog outputs.
There are a couple ways you could set this up, depending on your budget and the kind of interface you want.
You could get an interface with built-in low-latency mixing and monitoring. You would plug your microphone into your voice processor, guitar into pedals, and then plug the output of the voice processor into the interface and the output of the pedals into a separate input on the interface. Plug the outputs of the interface into a PA or other amplification system.
Then you would configure Live to use that interface. You would set the main mix for Live to be sent out of the interface outputs that are plugged into the PA, and you would configure two separate channels in Live with inputs from the interface - one channel configured to take audio from the guitar input, another channel for the vocal input.
Interfaces like this have control software that lets you mix the audio as it goes through the interface. You would set the mixing software to send a copy of the audio from each of the guitar and vocal inputs straight to the main outputs. That way, when you play or sing, it just comes right out of the PA with no delay. Based on the Live configuration, another copy of each signal goes into Live. You can use the APC40 to trigger recording and playback of vocal and guitar parts which will be mixed in with the vocal and guitar inputs on the mixer, and it will all come out of the PA at the same time. You do have to set up the channel monitoring settings in Live carefully for this configuration, so you don't have bad-sounding doubling of the audio when a track is record-armed.
Alternatively, you could get a low-latency interface. This will likely be the more expensive option, and might not be feasible for your computer. Very low latency round-trip audio can require a fast bus, like Thunderbolt, and a fast processor to process the audio quickly.
If you go this route, then you would set up the inputs and outputs in Live as above, but in this case you would configure the channel monitoring settings differently, so that when a track is not playing back a clip, it's processing the audio from the track input and then mixing it to the outputs inside Live. That means you can have effects in Live processing your guitar and voice to add to the effects you have already. You can record clips and playback clips as above.
In case this isn't already clear, you can also have Live generate additional audio by using virtual instrument tracks. For instance, you could put in a drum rack (that's a Live specific term for a drum virtual instrument) and play the pads on the APC40 to loop a drum beat, then you can play guitar and sing along with it. You can also set up bass and all kinds of other virtual instrument tracks, depending on what edition of Live you have and whether you've added any Live packs or other plug-ins to your setup.
And audio interface is a critical and central part of a setup like this, so it's important to choose wisely. They can get expensive, and as computer peripherals they can become obselete much faster than a guitar, for example, so buying the right interface can be a bit of an emotional journey.
Focusrite has been making some solid interfaces in recent years and they are worth a look. They have affordable options and high-end very low latency options.
Apogee makes very high end, high quality interfaces, expensive interfaces.
Univesal Audio is also fairly high end, and they include extra hardware processing to run plugins with low latency - this may be a bit overwhelming for someone just getting into computer audio, but you might keep this name in mind for later.
PreSonus and M-Audio make good quality, more affordable interfaces with a range of options.
MOTU has been around for years, making a range of interfaces, but I still don't know much about them.