Yes, the Canon T3i will take standard SD cards, no special purchase needed.
SD cards come in three types, SD, SDHC and SDXC. Nothing to worry about, since your camera will use any one of them. FYI, they just are capable of different amounts of data.
Now, you need to decide what size to get, 4GB, 8GB, 16GB, etc. If you are like me, you will download your pix at the end of the day and look at them on the computer and pick some to send to friends and family, so a 4GB or 8GB is plenty big. If you are one of those that will be showing off your shots to friends and family on the camera, you might want a larger card, such as the 16GB or 32GB. Keeping them on your camera is not recommended since the design was based around emptying the card at the end of every shoot, and reformatting it. Lots of people do keep their photos on their camera, at least for a while. If you have a smart phone or a tablet, you can transfer your favorite shots to that, and clean up the cameras memory card.
Next SD cards come in speed classes. The most common, and cheapest, is the Class 4, which is for most people. The only reason to get a higher class is if you are doing HD video. For HD video you will want a Class 6. For Full HD, you will want a Class 10, which is sometimes called UHS-1. So if you want to cover all the possibilities, get a Class 10/UHS-1 card.
As for brands, most of the name brands are good. Generic cards are bad. SanDisk and Fujifilm make good cards, I had a terrible experience with a Sony SD card and advise staying away from them.
Yes, you can get them at Amazon.com, or any online camera retailer, or any big box store that sells cameras. I get mine at Walmart.