When you're playing the game, hit the item menu. Go here and you'll notice something that you might not be expecting. There will be an item called the Everstone. When you have this item, that means that your Magikarp cannot evolve.

So what you want to do is to get rid of the Everstone. There's a little-known way to do that. Go to the pond where you feed your Magikarp at. Tap on your Magikarp over and over and over. You're going to need to do a lot of tapping in order to pull this off. Keep tapping and eventually, at random, the Everstone will break.

Once it does that, your Magikarp will then be able to evolve. You have to be at trainer rank 10 though in order to make it happen. Once you are at rank 10, your Magikarp will then have a maximum level of 20. Once you evolve it all the way up to level 20, it will automatically evolve into a Gyarados.

There is a downside to this, though. Currently, you cannot play with a Gyarados. Once it evolves, the mayor will take it away. Then you will be forced to start over again with a brand new Magikarp. Eventually, the developers might add an event to the game though that gives you the ability to compete with Gyarados, or even simply change the game around, but not right now.

To get a Red Gyarados, do all of the above steps, but fish out a Shiny Gold Magikarp first. These things are rare, and with the old rod, they are EXTREMELY rare; however, there are some tricks and cheats that you can do in order to get an easy one.