I would like to know the best programs that will allow me to backup my
Xbox games. I need to know if the program can be bought in stores,
online or both. I would also like to know price, but not necessary.
Ease and quickness is a must(I'm fairly computer literate, so it
doesn't have to be cake to use program). Three programs or more would
be nice as well. And just for the record, I know I need a modchip to
play backups.

I think you can use alcohol 120% to make copies of xbox games.

As for playing them, you don't necessarily have to use a modchip. 
Depending on which version of the xbox you have you can software mod
it.

Yes you can play backups without a modchip....no you can't make
backups without a modchip...so that doesn't help you too much. Go to
www.xbox-scene.com

You can get a modchip there as well as a solderless adapter to install
it. (Obviously keep in mind that opening the xbox voids the warranty)
Once you have installed the mod chip it will have a bios flashed to
it. You will want to flash it with another program called Slayer's
Evox. This adds a dashboard to the system. This will allow you to FTP
to your xbox. So you then hook your xbox up to your computer via a
crossover cable or through a switch. Once hooked up you need to FTP to
the xbox, grab the files off of the Disc, then you need to compress
the files into an XBE file on your computer. Then you need to burn
them to a DVD-r or DVD+R that is compatible with your xbox drive.
There are 3 different companies that made drives for the xbox so
finding a DVD brand that works with your xbox is harder than it
seems...Good luck...

You absolutely can make backups without a mod chip...you can make
backups with a software modded xbox.