To collect a Time Stone, you will need to finish a Special Stage. In order to get to a Special Stage, you will need to finish any Zone 1 or Zone 2 with at least 50 coins. Once you see the sign with Dr. Eggman giving a thumbs down on it, you should see a giant floating coin behind it. Jump through that coin. It will take you to a Special Stage.
I would highly recommend backing out of the game at this point in time, and copying your file to a memory stick or any other storage device. It is very, very easy to run out of time on the Special Stages, and when you do, the game saves automatically and pushes you to the next stage. If you copy your file to another storage device, at this point you can just back out of the game, recopy your copied save to the storage device that you play the game on, and play the Special Stage over again.
Once you start the Special Stage, you will notice there are a lot of flying UFOs. You will need to jump into all of the flying UFOs in order to destroy them and finish the stage. If your timer falls below twenty seconds, you will notice a new UFO in the middle of the stage with red lights. Go over to that UFO and destroy it to have your timer bumped up by 30 seconds. The timer UFO respawns every time your timer falls below 20 seconds, so you can continuously bump up your timer until you finish the level. You can get Time Stones on any Zone 1 or Zone 2 levels. There are 14 possible chances to get the 7 Time Stones, so don't fret if you miss a chance to get a Time Stone or two.