The desktop version is better than the web when you want more stability and quality. The desktop app uses less internet resources and keeps the connection more stable, while the browser can freeze up if you have too many tabs open. If your company offers it, just download the app from the official Skype website - it's easy to install.
To get decent audio and video, a few tweaks make a difference. Go to Settings > Audio and Video and test your microphone and camera before the actual meeting - Skype lets you do that right there. If your camera is bad, point it toward natural light or a lamp, because cameras hate low light. If your audio is choppy, check if other apps are using the microphone at the same time (sometimes antivirus or Chrome hog resources) and close them before you join the call.
Another thing nobody really talks about is testing your internet a few minutes before. If you're using Wi-Fi, stay close to the router or connect with a cable if you can - that's the safest way to not get disconnected in the middle of the meeting. And turn off notifications from your phone or other apps during the call so you don't have weird noise in the background. Do a test meeting with a trusted coworker to see how you come across before you get on with everyone - it makes you less nervous.