Choosing the best cheese for your grilled cheese sandwich is partly a question of personal preference. I definitely advise you to try out more variations than the traditional cheddar or American cheese.
Possibly the most common and traditional cheese to go with your grilled cheese is Cheddar or American cheese. There exist different varieties of these, but if you like the kind of fast food texture, American is definitely your choice.
More on the gourmet side you could use a sharp cheddar. This will give a much more interesting flavor profile to your grilled cheese, and obviously a sharper taste experience. Even mixing two varieties, e.g. half sharp with half mild cheddar, is a good idea in case you want to experiment with your grilled cheese sandwich.
This is definitely my favorite. Emmental is always an ideal choice for a grilled cheese–not too sharp, yet with a distinct taste. But Gruyère is also a nice option if you like stronger flavors.
Raclette is a less known Swiss variety, but it also melts nicely and is a very tasty type of cheese, definitely a go for your grilled cheese.
As I said, picking the best cheese for grilled cheese is really up to your choice. And I mean it. If you love cheese–I bet you do if you care this much about your grilled cheese sandwich, you should definitely give a shot at basically any type of cheese that melts easily.
I’ve seen grilled cheese sandwich recipes published with Brie, but even a good Camembert or some blue cheese would not be something that I would refuse to try.
And there is one last tip. Why not mix different types of cheese? If Gruyère is too strong for you, simply use half of it and the other half would be a milder Swiss cheese or Cheddar. You can also kick your grilled cheese up a notch with adding some grated hard cheese like Parmesan or Pecorino.