First off you will need a computer with at least 8gig of Ram for the speed.
You will also need a sequencing package like Cubase or the like. Just google and youtube reviews as there is so many to choose from. I bought software from ebay which cost £25 and its as good as Cubase. You get over 200 recording tracks complete with about 10 audio tracks (for guitar / microphone). Also its has a huge built in library of effects processors and a good sound canvas for your instruments, all the GM midi and GX midi voice and 128 midi channels.
Next you will need an anologue / digital audio / midi interface (a box) that you plug your microphone into or guitar and then sends the signals via usb to your sequencing package. As you already have a mixer, take the left and right out put channels from that into your new DA Box. Now you are ready to record your guitar and voice onto the computer.
I would also buy a new sound card. as expensive as you can buy with stereo channel outputs to go into an amp and speakers.
Also there are hundreds of sites on the web offering backing tracks for singing and guitar. or like me, I convert youtube video to mp3. Sear on youtube for say guitar backing tracks.
Drag the address into the Youtube/mp3 convertor and let it do its thing. Now you have an MP3 track for free on ya computer. Drag and drop the file into track no.1 on the sequescing package. Cut / crop / extend fade etc etc the way you want it, increase volume here, lower it there etc, then click only the track no.2 for recording. Track one will play, track 2 will record.
then track 3 and so on. You dont have to record the track in one hit, you can do bars at a time if you need to.
Finally add some reverb or effects against each track if you want as its plays. Finally do the final mix where you can save all the tracks into one master track for listening too etc.
This is just the basics and the jist of it. Hope it helps.