These steps got it working for me:

1. Install prelease (3.0.0-4) of vue-fontawesome, which is compatible with Vue 3, and the icon dependencies:

   npm i --save @fortawesome/vue-fontawesome@prerelease
   npm i --save @fortawesome/fontawesome-svg-core
   npm i --save @fortawesome/free-solid-svg-icons
   

2. In main.js, select the icons from @fortawesome/free-solid-svg-icons to load:

   import { library } from "@fortawesome/fontawesome-svg-core";
   import { faPhone } from "@fortawesome/free-solid-svg-icons";
   
   library.add(faPhone);
   

3. Globally register the font-awesome-icon component:

   import { FontAwesomeIcon } from "@fortawesome/vue-fontawesome";
   
   createApp(App)
     .component("font-awesome-icon", FontAwesomeIcon)
     .mount("#app");
   

4. In src/App.vue, use the component like this (note the icon name is phone, not faPhone):

   <!-- explicit style -->
   <font-awesome-icon :icon="['fas', 'phone']" />
   
   <!-- implicit style (fas is assumed) -->
   <font-awesome-icon icon="phone" />
   

demo

Tony19 is correct but if you want to avoid cluttering your main.ts(js) file, you can do something like this:

What you can do is create a separate file:

fontawesome-icons.ts

import { FontAwesomeIcon } from "@fortawesome/vue-fontawesome";
import { library } from "@fortawesome/fontawesome-svg-core";
import { faPhone } from "@fortawesome/free-solid-svg-icons";
import { faUser } from "@fortawesome/free-solid-svg-icons";
import { faFlag } from "@fortawesome/free-solid-svg-icons";

library.add(faPhone);
library.add(faUser);
library.add(faFlag);

export default FontAwesomeIcon;

Then in your main.ts:

import { createApp } from "vue";
import App from "./App.vue";
import "./registerServiceWorker";
import "@/assets/css/main.scss";
import router from "./router";
import store from "./store";
import FontAwesomeIcon from "@/utilities/fontawesome";

createApp(App)
    .component("font-awesome-icon", FontAwesomeIcon)
    .use(store)
    .use(router)
    .mount("#app");

for ease but compromising on bundle size and speed. you can import all icons at once

import { library } from '@fortawesome/fontawesome-svg-core'
import { fas } from '@fortawesome/free-solid-svg-icons'
import { far } from '@fortawesome/free-regular-svg-icons'
import { fab } from '@fortawesome/free-brands-svg-icons'

//Add all icons to the library so you can use it in your page
library.add(fas, far, fab)

source official fontawesome docs

Install

npm i --save @fortawesome/fontawesome-svg-core
npm i --save @fortawesome/free-solid-svg-icons
npm i --save @fortawesome/free-regular-svg-icons
npm i --save @fortawesome/free-brands-svg-icons

On your main.ts

import { library, dom } from "@fortawesome/fontawesome-svg-core";
import { FontAwesomeIcon } from "@fortawesome/vue-fontawesome";
import { fas } from '@fortawesome/free-solid-svg-icons'
import { fab } from '@fortawesome/free-brands-svg-icons';
import { far } from '@fortawesome/free-regular-svg-icons';
library.add(fas, far, fab)
dom.watch();

createApp(App)
    .component("font-awesome-icon", FontAwesomeIcon)
    .mount('#app')

To use on .vue files

<i class="fa-solid fa-arrow-left"></i>
<i class="fa-brands fa-facebook"></i>