The guitar amps you have most likely have some "effects" on them already. Most guitar amps allow you to select either a clean signal or a distorted overdrive signal (at living room levels). Also, many have built in reverb or delays or other effects that can be selected.
But eventually, most guitar players end up wanting a sound they just can't get out of their amp. So they buy a "stomp box" or pedal to add a different effect. You can get individual boxes (aka pedals) to add things such as tremolo, chorus, delay, octaves, volume boost, fuzz and many others.
Many performing guitarist will connect several of these pedals together with short cables (chaining one to the other) and attach them to what they call a "pedal board". Sometimes you can use more than one effect at the same time.
With a pedal board loaded with multiple effects pedals, you would plug your guitar into the first one and the signal would go through each of the other boxes in the chain and a different instrument cable will run from the last box in the chain into the guitar amplifier.
If you want to use many different effects to play at home for yourself and your friends, a multi-effects processor may be the more economical way to go. There are many such processors on the market and for home use, you could use either a rack mount processor such as the Fractal Audio AXE FX (expensive) or a pedal/foot-switch controlled unit such as the Boss ME.
A digital multi-effects processor will allow you to plug your guitar into the unit and then run a cable from the processor to a PA (public address) system and whatever effects you select (overdrive, reverb, flanger etc.) will come through the PA speakers. Many of these processing units should be capable of allowing you to play your guitar through your home stereo as well but I would check with tech support from the manufacturer if it is not clear from the owners manual what type of signal the unit sends.
The disadvantage of many multi-effects processors, is that to get to a certain effect, you must bank up or down (tap the up or down button or turn a knob to scroll through a menu on a tiny screen) to find the effect you want. With a pedal board that consist of multiple single effects pedals, you just stomp the one you want and get it when you want it without scrolling through all the selections.
The biggest advantage of a multi-effects processor, is that you might get 30 or more different select-able effects. Also, many of these units offer amp cabinet emulation to make your guitar sound like you are playing through a 1965 Fender Twin Reverb or a Blues Junior or a Mesa Boogie guitar amp - only the sound is coming out of your PA speakers.
You can plug these multi effects processors into your guitar amp using a standard instrument cable and you should get excellent results. Use your clean channel setting and play around with the EQ controls on your amp and the adjustable parameters on the effects processor to tweak the sound the way you like it. Most units allow you to save your adjustments as a preset and then call up the preset on the menu later.
You can also plug the multi effects processor into your PA system and many of them allow you to send the signal to both a mixer to run into your PA as well as a guitar amp at the same time.
One thing that you might find extremely useful for what you want to do, is the new iRig Stomp iRig Stomp multi effects pedal This processor is a stage ready stompbox multi effects guitar interface that works with iPod, iPhone, or iPad and most iOS guitar apps including the popular AmpliTube App. It is inexpensive, has good reviews, and you can plug it into your guitar amp, a PA OR even your home stereo. You might need to buy an iPod to go with it if you don't have an iPhone or iPad.
To plug the iRig Stomp into your stereo, you need a cable like the dual quarter inch L/R to dual RCA L/R pictured below. Plug the 1/4 inch plugs into the left and right output on the pedal and the RCA left and right jacks into your auxiliary input jacks on your home stereo. Hope this helps and good luck.