For a baking beginner, what cake icing you recommend?

Buttercream is pretty simple, start with room temperature butter, beat it until it's white, slowly add icing sugar, add small amounts of milk as needed to keep the consistency right.

For each 1/4 cup (1/2 stick) of butter, use about 1 1/2 cups of icing sugar (sifted) and 3 Tbsp of milk. You can do it in a mixer, a food processor, or just use a bowl and a whisk. I used a whisk a few days ago when I realized my large stand mixer is overkill for the small amount of icing I wanted and I found that microwaving the icing for about 10 seconds helped me stir in the sugar when it was getting too thick.

Half the time when I'm making buttercream type frosting, I go with melted butter, and just mix icing sugar in to taste - forming a kind of paste or glaze consistency. It has a flatter, less fluffy consistency this way, instead of doing it the "right" way with whipped butter - but I am usually more concerned about the flavor profile, especially if its for casual eating rather than for guests. Its quick and easy and foolproof, this way, and easy to add make just a little or add more if running low, since you don't have to fluff ingredients separately.

Other than that, you can make a glaze quickly by making a thick liquid or paste of the powdered sugar with any reasonable liquid (or, yanno, unreasonable ones if that floats your boat). Water is easy, as ann mentioned, but you can use milk, or juice if you want a quickly flavored glaze, melted chocolate, any syrup that strikes your fancy, whatever.

There are different types of buttercream, but the one most commonly used by new bakers is American buttercream. It is made by mixing butter and powdered/confectioners sugar. Sounds like that's what you've made and what the other posters are recommending. American buttercream is a crusting buttercream. So the crust you mentioned is normal. If you prefer that it doesn't crust, you may want to try a different type of frosting or buttercream. Chocolate ganache is a great frosting and very easy. Use 1:1 heavy cream to chocolate that is chopped or chocolate chips. Heat the cream to almost boiling. Pour the hot cream over the chocolate. Let it sit for a few minutes. Then stir until smooth. Let it cool until you like the consistency. If you want, you can cool it all the way then whip it up for a lighter texture. It will not crust.

The most basic recipe is to mix icing sugar with a very small amount of water to make a paste. Drizzle it over the muffins. But if you want something that you can pipe then I agree buttercream is the way to go, I use this one... https://www.howtocookthat.net/public_html/buttercream-cupcake-frosting-recipes/