You can hook up an amp to the OEM head unit by using the OEM speaker wires for inputs.
(You don’t have to use a processor to hook up an amp to the factory head unit. There are processors from at least half a dozen manufacturers, and one of them may be right for your particular setup, to provide an improvement for what you want to do with your audio system - if you want to spend extra bucks. You can investigate and add one later if you think you need it.)
You need either a Line Output Converter (LOC) (about $15 for a Scosche 2 channel at Crutchfield) and regular amp, or an amplifier that will accept high or line level inputs (i.e. built-in LOC). There are a good number of choices with high or line level inputs, and if you want to find some quick, check out the Crutchfield catalog amp comparison chart. Some amps require some speaker-wire-to-RCA adapters (Crutchfield sells a JL Audio 2 channel adapter for about $10).
I agree with others that using stock speakers with an add-on amp will probably end up destroying the stock speakers, so you will want to replace them, at least the front components. To replace the fronts and rears you need to be looking at 4 channel amps (for simpler hookup choosing those with line level inputs). For the front speakers I would recommend you go to the Crutchfield website and see which 6.5" component speakers fit your car, watching out of how deep they go into the door from surface mounting. Then you can determine a maximum speaker depth, to go shopping around for other speaker selections. (Maybe someone here already knows the maximum speaker depth of the 10th generation Corollas from experience.)
If you get a bridgeable 4/3/2 four channel amp with enough power, you can always disconnect the rear fill speakers and bridge the rear channels to drive a trunk subwoofer in the future, if you want deep bass.
The amplifier power output ratings can be deceptively high on some manufacturers stated specs. I look at the power ratings at 4 ohms that typically seem to be rated at around 14 volts. It is the ratings at about 12 volts that are about the most conservative. The Cadence amps are rated in both voltage ranges, so you can see the conservative 12 Volt power is about 20 or 30 percent lower than 14 Volt ratings.
So I would recommend looking at the conservative ratings to more closely compare amplifier power to match your speaker requirements.
Another thing to look at is the amp fuse rating, for example. An amplifier that only requires 20 Amps is not going to have the power potential of an amp rated at 80 amps no matter what the manufacturers’ state as power output. There are other ratings, such as low THD (<1%) that are sometimes provided that are also important, but sizing up the amps real power potential is the first screening factor for me.
Also, here is a handy site for choosing sellers that don’t abuse their customers with fraudulent practices.
http://www.resellerratings.com/store/onlinecarstereo
Here are a couple of well-rated sellers to start with. Crutchfield (internet) has loads of information and help and is typically close to retail pricing. Woofers, etc. (internet) is a well-rated deep-discount seller with a wide variety of brands and items.
http://www.crutchfield.com/
http://www.woofersetc.com/